Physics · Article
Critical phenomena, scaling, and renormalization
Begin with an exact spin-chain coarse-graining, distinguish a changed ruler from a changed physical correlation length, then derive fixed-point scaling, Gaussian limits and the conditions on universality and critical exponents.
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Back to PhysicsAfter reading, you can
- Define a connected correlation and distinguish a physical correlation length from its value in rescaled lattice units.
- Derive an exact one-dimensional decimation including the partition-sum normalization and its zero-temperature limit.
- Separate summing short-scale degrees of freedom, coordinate rescaling and field normalization.
- Classify relevant, irrelevant and marginal perturbations under a stated flow convention and explain universality with its conditions.
- Derive conventional critical-exponent relations and Gaussian engineering dimensions without applying them outside their regime.
- Explain dangerously irrelevant variables, finite-size corrections and why static scaling does not determine dynamics.
One chain, two choices of ruler
Imagine a very long straight chain of two-valued spins, each +1 or −1, separated by a distance a. In the supplied equilibrium example, the correlation of two spins r sites apart is C(r)=(1/2)^r. Neighbours have correlation 1/2; spins two sites apart have correlation 1/4. Now keep every other spin and sum over the spins between them. Have correlations become shorter in physical space, or have we changed the ruler?
The retained neighbours were two old sites apart, so their correlation is now 1/4. An exponential correlation length is the distance at which the exponential falls by a factor e. It is a/ln2, about 1.4426950a, before the operation. Measured in new lattice spacings it is only 0.72134752, but each new spacing is 2a. The physical length has not changed. This distinction is the first step toward understanding a renormalization-group calculation.
The chain is an exact, deliberately simple equilibrium model. A spin need not be a literal tiny rotating magnet; here it is a variable with two allowed values. We will derive its correlation and the precise sum-over-spins operation, then use separate models to explain critical scaling. The chain itself has no transition at a positive temperature.
Follow a first reading and an advanced return
On a first reading, follow the chain, the physical correlation length, the three coarse-graining steps, and the growing or shrinking perturbations around a fixed point. Correlation means an equilibrium statistical relationship, not two particles sending a signal instantaneously. A fixed point is a statistical description unchanged by the stated change of scale, not a sample frozen in time.
The advanced return path derives scaling of the singular free energy, critical exponents, Gaussian engineering dimensions, the upper critical dimension, and the reason an apparently irrelevant interaction sometimes cannot be discarded. Published probability, ensemble and partition-function lessons provide preparation. Necessary meanings are introduced here; the phase-transition topic supplies the adjacent discussion of order parameters and competing minima.
This lesson is about static critical behaviour and descriptions at different spatial resolutions. Transport, equilibration rates, stochastic dynamics, detailed quantum field-theory renormalization and a full catalogue of universality classes have their own permanent subjects. Keeping those connections does not require treating every such theory as part of the same example.
Measure connected correlations
For a local variable s_i, the connected correlation is G(i,j)=⟨s_i s_j⟩−⟨s_i⟩⟨s_j⟩. Subtracting the product of the means removes the part explained by a uniform average alone. An ordered state can have a nonzero mean; that is why an unconnected product is not always the quantity whose decay defines a fluctuation length.
In our infinite zero-field chain at every finite positive temperature, the mean spin is zero. Its connected and unconnected two-point correlations are therefore equal: G(r)=C(r)=q^r, with q=tanhK, K=J/(k_B T), and ferromagnetic coupling J>0. The physical separation is ra. Writing q^r=exp(−ra/ξ) gives ξ/a=−1/lnq for 0<q<1. This is a spatial equilibrium correlation, not a time trace or probability density.
One way to derive C(r) is to use the zero-field conditional expectation of a neighbouring spin, ⟨s_(i+1)|s_i⟩=q s_i. Repeating that conditional averaging r times gives ⟨s_r|s_0⟩=q^r s_0. Since s_0²=1, averaging its product with s_0 gives q^r. This derivation uses the infinite homogeneous nearest-neighbour chain; a small periodic ring has finite-size corrections.
Near an ordinary continuous transition in higher dimensions, connected correlations often have an algebraic factor as well as a long-distance cutoff. At criticality, a short-range isotropic system in its scaling regime has G(R) proportional to R^(−d+2−η), with spatial dimension d and anomalous exponent η. Away from criticality a function of R/ξ cuts off that behaviour. Microscopic separations, anisotropy, long-range forces and boundaries can require a different description.
Sum a middle spin without throwing away its weight
Take two retained endpoint spins s_L and s_R with a middle spin s. The original statistical weight of this short piece is exp[K(s_L s+s s_R)]. Summing over s=+1 and s=−1 produces 2cosh[K(s_L+s_R)]. This is a sum over equilibrium alternatives, not a rule that sets the middle spin equal to its average.
We want to express that sum as A exp(K′ s_L s_R). For aligned endpoints it must equal 2cosh2K; for opposed endpoints it must equal 2. Thus A expK′=2cosh2K and A exp(−K′)=2. Dividing the equations gives K′=(1/2)lncosh2K; multiplying them gives A=2√cosh2K. Both the new coupling and the normalization are part of the exact identity.
For a periodic chain with even N, this operation on every other spin gives Z_N(K)=A^(N/2) Z_(N/2)(K′). Dropping A leaves ratios of probabilities for retained configurations unchanged, but loses the corresponding contribution to lnZ and the free energy. Because A depends on K, it cannot simply be ignored when computing thermodynamic derivatives.
The nearest-neighbour form closes exactly for this zero-field one-dimensional chain. In a more general model, summing short-scale variables generates further interactions consistent with the symmetries. Keeping only a few couplings then becomes an approximation. A diagram of a few arrows in coupling space does not by itself establish that this truncation is accurate.
Change the lattice spacing and keep the physical length
The exact transformation is particularly transparent in q=tanhK: q′=q². After n operations, q_n=q_0^(2^n) and the retained lattice spacing is a_n=2^n a. The correlation length counted in retained sites is −1/lnq_n=(−1/lnq_0)/2^n. Multiplying by a_n returns the original physical ξ.
For the opening q_0=1/2, the retained correlations become 1/4, 1/16 and 1/256 on successive steps. Those are correlations across physical separations 2a, 4a and 8a. They are not repeated measurements across the same original nearest-neighbour bond. The table shows why the number of sites and the physical length must be kept separate.
A renormalization-group convention often goes one step further and rescales coordinates so that the new microscopic spacing is again called one unit. In those rescaled coordinates ξ′=ξ/2. There is no contradiction with conservation of the original physical length: one statement uses the new coordinate unit, and the other restores that unit in metres or in the original a.
Keep the physical ruler; retain the statistical weight
Ask what changes when alternate spins are removed: site labels and coupling change, but an original physical separation must still be measured with the same ruler.
Same positions, different site counts
- Original sites
- Retained even-indexed sites
The two removed-spin alternatives add
- Removed spin +1
- Removed spin −1
- The ruler and the sum over alternatives describe an equilibrium transformation, not motion of a spin or elapsed time.
- The local normalization A is part of the exact identity. It cancels in retained probability ratios, but contributes to the partition sum and free energy.
The original whole-table links and complete additional values are below.
Exact local weights for each retained endpoint pair. The removed spin has two alternatives whose weights add; these are not normalized probabilities.
| q₀ | sL | sR | Weight for s=+1 | Weight for s=−1 | Sum over s |
|---|---|---|---|---|---|
| 1/5 | 1 | 1 | 3/2 | 2/3 | 13/6 |
| 1/5 | 1 | -1 | 1 | 1 | 2 |
| 1/5 | -1 | 1 | 1 | 1 | 2 |
| 1/5 | -1 | -1 | 2/3 | 3/2 | 13/6 |
| 1/2 | 1 | 1 | 3 | 1/3 | 10/3 |
| 1/2 | 1 | -1 | 1 | 1 | 2 |
| 1/2 | -1 | 1 | 1 | 1 | 2 |
| 1/2 | -1 | -1 | 1/3 | 3 | 10/3 |
| 4/5 | 1 | 1 | 9 | 1/9 | 82/9 |
| 4/5 | 1 | -1 | 1 | 1 | 2 |
| 4/5 | -1 | 1 | 1 | 1 | 2 |
| 4/5 | -1 | -1 | 1/9 | 9 | 82/9 |
| 19/20 | 1 | 1 | 39 | 1/39 | 1522/39 |
| 19/20 | 1 | -1 | 1 | 1 | 2 |
| 19/20 | -1 | 1 | 1 | 1 | 2 |
| 19/20 | -1 | -1 | 1/39 | 39 | 1522/39 |
| 99/100 | 1 | 1 | 199 | 1/199 | 39602/199 |
| 99/100 | 1 | -1 | 1 | 1 | 2 |
| 99/100 | -1 | 1 | 1 | 1 | 2 |
| 99/100 | -1 | -1 | 1/199 | 199 | 39602/199 |
Complete periodic 8-to-4 partition identities. Every original 256-state sum and retained 16-state sum is included through its exact rational value.
| q₀ | Original states | Retained states | A⁴ (exact) | Z₈ (exact) | Z₄(K′) (exact) |
|---|---|---|---|---|---|
| 1/5 | 256 | 16 | 169/9 | 195313/648 | 195313/12168 |
| 1/2 | 256 | 16 | 400/9 | 65792/81 | 4112/225 |
| 4/5 | 256 | 16 | 26896/81 | 116777216/6561 | 7298576/136161 |
| 19/20 | 256 | 16 | 9265936/1521 | 10901392138496/2313441 | 681337008656/880843041 |
| 99/100 | 256 | 16 | 6273273616/39601 | 4922226417735475456/1568239201 | 307639151108467216/15526744279201 |
On a narrow screen, focus or swipe within a figure or table to scroll horizontally. Labels retain their designed size.
| Step n | q_n | a_n/a | ξ in retained sites | Physical ξ/a |
|---|---|---|---|---|
| 0 | 1/2 | 1 | 1/ln2 | 1/ln2 |
| 1 | 1/4 | 2 | 1/(2ln2) | 1/ln2 |
| 2 | 1/16 | 4 | 1/(4ln2) | 1/ln2 |
| 3 | 1/256 | 8 | 1/(8ln2) | 1/ln2 |
Recognize what this chain can and cannot teach
For every 0<q<1, squaring q makes it smaller. Repeated decimation reaches the high-temperature fixed point q=0. The boundary q=1 corresponds to K→∞ and T→0; it is the other fixed point. There is no intermediate positive-temperature critical point in this infinite nearest-neighbour chain.
Near q=1, put δ=1−q. The exact recursion is δ′=2δ−δ², so a small δ grows approximately by a factor 2 under a change of length scale by 2. This is a relevant perturbation in the declared coupling coordinate. It is not a proof that ξ is proportional to T^(−1). In fact, ξ/a=−1/ln[tanh(J/(k_B T))] is asymptotic to (1/2)exp[2J/(k_B T)] as T approaches zero from above.
A reduced temperature t=(T−T_c)/T_c is useful for a positive T_c, but it is undefined when T_c=0. The chain shows exact coarse-graining and a diverging zero-temperature length without supplying the ordinary finite-temperature power law. At q=0 use the limiting ξ=0 rather than evaluating ln0; at q=1 keep ξ infinite rather than inserting an invented large finite number.
Follow a spatial scale step, not a clock
Compare the exact q map with the spatial correlations it describes. The initial and sixth-step plots use different, explicitly labelled vertical windows because one retained bond spans 64 original spacings after six steps.
Exact map and its two endpoints
- Exact rule q′=q²
- Reference q′=q
- Opening q₀=1/2: six spatial scale steps
Initial chain: all five controls
- q₀=1/5
- q₀=1/2
- q₀=4/5
- q₀=19/20
- q₀=99/100
Sixth step: keep the very small positive tails
- q₀=1/5
- q₀=1/2
- q₀=4/5
- q₀=19/20
- q₀=99/100
- The fixed point q=1 is the T=0 boundary, not a positive critical temperature. The q=0 endpoint is a limit; no logarithm of zero is evaluated.
- The two correlation panels have ordinate windows −45…0 and −3000…0, respectively. They are not endpoint-normalized copies of a curve.
- Only the initial and sixth-step curves are plotted here. Every intermediate assigned step and every r=0…64 value is retained below, including 241 positive values below ordinary floating-point range.
- Markers are the discrete samples; connecting segments guide the eye. Neither the iteration index nor spatial decay is a relaxation time.
The original whole-table links and complete additional values are below.
Every assigned decimation control. Numerical values are rounded to 12 significant digits; the source model and formulas define the exact quantities.
| q₀ | Step n | qₙ≈ | Kₙ≈ | K′ₙ≈ | Aₙ≈ | ξ in retained sites≈ | aₙ/a | Physical ξ/a≈ |
|---|---|---|---|---|---|---|---|---|
| 1/5 | 0 | 2.00000000000e-1 | 2.02732554054e-1 | 4.00213538368e-2 | 2.08166599947e+0 | 6.21334934560e-1 | 1 | 6.21334934560e-1 |
| 1/5 | 1 | 4.00000000000e-2 | 4.00213538368e-2 | 1.60000136534e-3 | 2.00320256410e+0 | 3.10667467280e-1 | 2 | 6.21334934560e-1 |
| 1/5 | 2 | 1.60000000000e-3 | 1.60000136534e-3 | 2.56000000001e-6 | 2.00000512001e+0 | 1.55333733640e-1 | 4 | 6.21334934560e-1 |
| 1/5 | 3 | 2.56000000000e-6 | 2.56000000001e-6 | 6.55360000000e-12 | 2.00000000001e+0 | 7.76668668200e-2 | 8 | 6.21334934560e-1 |
| 1/5 | 4 | 6.55360000000e-12 | 6.55360000000e-12 | 4.29496729600e-23 | 2.00000000000e+0 | 3.88334334100e-2 | 16 | 6.21334934560e-1 |
| 1/5 | 5 | 4.29496729600e-23 | 4.29496729600e-23 | 1.84467440737e-45 | 2.00000000000e+0 | 1.94167167050e-2 | 32 | 6.21334934560e-1 |
| 1/5 | 6 | 1.84467440737e-45 | 1.84467440737e-45 | 3.40282366921e-90 | 2.00000000000e+0 | 9.70835835249e-3 | 64 | 6.21334934560e-1 |
| 1/2 | 0 | 5.00000000000e-1 | 5.49306144334e-1 | 2.55412811883e-1 | 2.58198889747e+0 | 1.44269504089e+0 | 1 | 1.44269504089e+0 |
| 1/2 | 1 | 2.50000000000e-1 | 2.55412811883e-1 | 6.25815714770e-2 | 2.12916258969e+0 | 7.21347520444e-1 | 2 | 1.44269504089e+0 |
| 1/2 | 2 | 6.25000000000e-2 | 6.25815714770e-2 | 3.90626986840e-3 | 2.00782781857e+0 | 3.60673760222e-1 | 4 | 1.44269504089e+0 |
| 1/2 | 3 | 3.90625000000e-3 | 3.90626986840e-3 | 1.52587890637e-5 | 2.00003051781e+0 | 1.80336880111e-1 | 8 | 1.44269504089e+0 |
| 1/2 | 4 | 1.52587890625e-5 | 1.52587890637e-5 | 2.32830643654e-10 | 2.00000000047e+0 | 9.01684400556e-2 | 16 | 1.44269504089e+0 |
| 1/2 | 5 | 2.32830643654e-10 | 2.32830643654e-10 | 5.42101086243e-20 | 2.00000000000e+0 | 4.50842200278e-2 | 32 | 1.44269504089e+0 |
| 1/2 | 6 | 5.42101086243e-20 | 5.42101086243e-20 | 2.93873587706e-39 | 2.00000000000e+0 | 2.25421100139e-2 | 64 | 1.44269504089e+0 |
| 4/5 | 0 | 8.00000000000e-1 | 1.09861228867e+0 | 7.58173744684e-1 | 4.26874949162e+0 | 4.48142011772e+0 | 1 | 4.48142011772e+0 |
| 4/5 | 1 | 6.40000000000e-1 | 7.58173744684e-1 | 4.35130490948e-1 | 3.09032935160e+0 | 2.24071005886e+0 | 2 | 4.48142011772e+0 |
| 4/5 | 2 | 4.09600000000e-1 | 4.35130490948e-1 | 1.69373413190e-1 | 2.36912477514e+0 | 1.12035502943e+0 | 4 | 4.48142011772e+0 |
| 4/5 | 3 | 1.67772160000e-1 | 1.69373413190e-1 | 2.81549347885e-2 | 2.05711006207e+0 | 5.60177514716e-1 | 8 | 4.48142011772e+0 |
| 4/5 | 4 | 2.81474976711e-2 | 2.81549347885e-2 | 7.92281790917e-4 | 2.00158519146e+0 | 2.80088757358e-1 | 16 | 4.48142011772e+0 |
| 4/5 | 5 | 7.92281625143e-4 | 7.92281790917e-4 | 6.27710173539e-7 | 2.00000125542e+0 | 1.40044378679e-1 | 32 | 4.48142011772e+0 |
| 4/5 | 6 | 6.27710173539e-7 | 6.27710173539e-7 | 3.94020061964e-13 | 2.00000000000e+0 | 7.00221893394e-2 | 64 | 4.48142011772e+0 |
| 19/20 | 0 | 9.50000000000e-1 | 1.83178082306e+0 | 1.48553585587e+0 | 8.83466366373e+0 | 1.94957257462e+1 | 1 | 1.94957257462e+1 |
| 19/20 | 1 | 9.02500000000e-1 | 1.48553585587e+0 | 1.14027374118e+0 | 6.25524881523e+0 | 9.74786287311e+0 | 2 | 1.94957257462e+1 |
| 19/20 | 2 | 8.14506250000e-1 | 1.14027374118e+0 | 7.98898340655e-1 | 4.44618098110e+0 | 4.87393143656e+0 | 4 | 1.94957257462e+1 |
| 19/20 | 3 | 6.63420431289e-1 | 7.98898340655e-1 | 4.72387894460e-1 | 3.20763875147e+0 | 2.43696571828e+0 | 8 | 1.94957257462e+1 |
| 19/20 | 4 | 4.40126668652e-1 | 4.72387894460e-1 | 1.96190494382e-1 | 2.43351733790e+0 | 1.21848285914e+0 | 16 | 1.94957257462e+1 |
| 19/20 | 5 | 1.93711484459e-1 | 1.96190494382e-1 | 3.75417661981e-2 | 2.07651072031e+0 | 6.09241429569e-1 | 32 | 1.94957257462e+1 |
| 19/20 | 6 | 3.75241392111e-2 | 3.75417661981e-2 | 1.40806195409e-3 | 2.00281810748e+0 | 3.04620714785e-1 | 64 | 1.94957257462e+1 |
| 99/100 | 0 | 9.90000000000e-1 | 2.64665241236e+0 | 2.30009144787e+0 | 1.99501892285e+1 | 9.94991624734e+1 | 1 | 9.94991624734e+1 |
| 99/100 | 1 | 9.80100000000e-1 | 2.30009144787e+0 | 1.95356835626e+0 | 1.41076264878e+1 | 4.97495812367e+1 | 2 | 9.94991624734e+1 |
| 99/100 | 2 | 9.60596010000e-1 | 1.95356835626e+0 | 1.60719668930e+0 | 9.97761286533e+0 | 2.48747906184e+1 | 4 | 9.94991624734e+1 |
| 99/100 | 3 | 9.22744694428e-1 | 1.60719668930e+0 | 1.26142965230e+0 | 7.06093043768e+0 | 1.24373953092e+1 | 8 | 9.94991624734e+1 |
| 99/100 | 4 | 8.51457771095e-1 | 1.26142965230e+0 | 9.18064164619e-1 | 5.00887503103e+0 | 6.21869765459e+0 | 16 | 9.94991624734e+1 |
| 99/100 | 5 | 7.24980335958e-1 | 9.18064164619e-1 | 5.84041250118e-1 | 3.58654172592e+0 | 3.10934882729e+0 | 32 | 9.94991624734e+1 |
| 99/100 | 6 | 5.25596487526e-1 | 5.84041250118e-1 | 2.83619475780e-1 | 2.65585505203e+0 | 1.55467441365e+0 | 64 | 9.94991624734e+1 |
Complete spatial correlations for q₀=1/5: all seven steps and 65 retained-site distances. Scientific values are rounded to 12 significant digits; the final column is exact. No positive tail is replaced by zero.
| Step n | r in retained sites | Physical R/a=2ⁿr | Cₙ(r)≈ | Exact Cₙ(r) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 0 | 1 | 1 | 2.00000000000e-1 | (1/5)^1 |
| 0 | 2 | 2 | 4.00000000000e-2 | (1/5)^2 |
| 0 | 3 | 3 | 8.00000000000e-3 | (1/5)^3 |
| 0 | 4 | 4 | 1.60000000000e-3 | (1/5)^4 |
| 0 | 5 | 5 | 3.20000000000e-4 | (1/5)^5 |
| 0 | 6 | 6 | 6.40000000000e-5 | (1/5)^6 |
| 0 | 7 | 7 | 1.28000000000e-5 | (1/5)^7 |
| 0 | 8 | 8 | 2.56000000000e-6 | (1/5)^8 |
| 0 | 9 | 9 | 5.12000000000e-7 | (1/5)^9 |
| 0 | 10 | 10 | 1.02400000000e-7 | (1/5)^10 |
| 0 | 11 | 11 | 2.04800000000e-8 | (1/5)^11 |
| 0 | 12 | 12 | 4.09600000000e-9 | (1/5)^12 |
| 0 | 13 | 13 | 8.19200000000e-10 | (1/5)^13 |
| 0 | 14 | 14 | 1.63840000000e-10 | (1/5)^14 |
| 0 | 15 | 15 | 3.27680000000e-11 | (1/5)^15 |
| 0 | 16 | 16 | 6.55360000000e-12 | (1/5)^16 |
| 0 | 17 | 17 | 1.31072000000e-12 | (1/5)^17 |
| 0 | 18 | 18 | 2.62144000000e-13 | (1/5)^18 |
| 0 | 19 | 19 | 5.24288000000e-14 | (1/5)^19 |
| 0 | 20 | 20 | 1.04857600000e-14 | (1/5)^20 |
| 0 | 21 | 21 | 2.09715200000e-15 | (1/5)^21 |
| 0 | 22 | 22 | 4.19430400000e-16 | (1/5)^22 |
| 0 | 23 | 23 | 8.38860800000e-17 | (1/5)^23 |
| 0 | 24 | 24 | 1.67772160000e-17 | (1/5)^24 |
| 0 | 25 | 25 | 3.35544320000e-18 | (1/5)^25 |
| 0 | 26 | 26 | 6.71088640000e-19 | (1/5)^26 |
| 0 | 27 | 27 | 1.34217728000e-19 | (1/5)^27 |
| 0 | 28 | 28 | 2.68435456000e-20 | (1/5)^28 |
| 0 | 29 | 29 | 5.36870912000e-21 | (1/5)^29 |
| 0 | 30 | 30 | 1.07374182400e-21 | (1/5)^30 |
| 0 | 31 | 31 | 2.14748364800e-22 | (1/5)^31 |
| 0 | 32 | 32 | 4.29496729600e-23 | (1/5)^32 |
| 0 | 33 | 33 | 8.58993459200e-24 | (1/5)^33 |
| 0 | 34 | 34 | 1.71798691840e-24 | (1/5)^34 |
| 0 | 35 | 35 | 3.43597383680e-25 | (1/5)^35 |
| 0 | 36 | 36 | 6.87194767360e-26 | (1/5)^36 |
| 0 | 37 | 37 | 1.37438953472e-26 | (1/5)^37 |
| 0 | 38 | 38 | 2.74877906944e-27 | (1/5)^38 |
| 0 | 39 | 39 | 5.49755813888e-28 | (1/5)^39 |
| 0 | 40 | 40 | 1.09951162778e-28 | (1/5)^40 |
| 0 | 41 | 41 | 2.19902325555e-29 | (1/5)^41 |
| 0 | 42 | 42 | 4.39804651110e-30 | (1/5)^42 |
| 0 | 43 | 43 | 8.79609302221e-31 | (1/5)^43 |
| 0 | 44 | 44 | 1.75921860444e-31 | (1/5)^44 |
| 0 | 45 | 45 | 3.51843720888e-32 | (1/5)^45 |
| 0 | 46 | 46 | 7.03687441777e-33 | (1/5)^46 |
| 0 | 47 | 47 | 1.40737488355e-33 | (1/5)^47 |
| 0 | 48 | 48 | 2.81474976711e-34 | (1/5)^48 |
| 0 | 49 | 49 | 5.62949953421e-35 | (1/5)^49 |
| 0 | 50 | 50 | 1.12589990684e-35 | (1/5)^50 |
| 0 | 51 | 51 | 2.25179981369e-36 | (1/5)^51 |
| 0 | 52 | 52 | 4.50359962737e-37 | (1/5)^52 |
| 0 | 53 | 53 | 9.00719925474e-38 | (1/5)^53 |
| 0 | 54 | 54 | 1.80143985095e-38 | (1/5)^54 |
| 0 | 55 | 55 | 3.60287970190e-39 | (1/5)^55 |
| 0 | 56 | 56 | 7.20575940379e-40 | (1/5)^56 |
| 0 | 57 | 57 | 1.44115188076e-40 | (1/5)^57 |
| 0 | 58 | 58 | 2.88230376152e-41 | (1/5)^58 |
| 0 | 59 | 59 | 5.76460752303e-42 | (1/5)^59 |
| 0 | 60 | 60 | 1.15292150461e-42 | (1/5)^60 |
| 0 | 61 | 61 | 2.30584300921e-43 | (1/5)^61 |
| 0 | 62 | 62 | 4.61168601843e-44 | (1/5)^62 |
| 0 | 63 | 63 | 9.22337203685e-45 | (1/5)^63 |
| 0 | 64 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 1 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 1 | 1 | 2 | 4.00000000000e-2 | (1/5)^2 |
| 1 | 2 | 4 | 1.60000000000e-3 | (1/5)^4 |
| 1 | 3 | 6 | 6.40000000000e-5 | (1/5)^6 |
| 1 | 4 | 8 | 2.56000000000e-6 | (1/5)^8 |
| 1 | 5 | 10 | 1.02400000000e-7 | (1/5)^10 |
| 1 | 6 | 12 | 4.09600000000e-9 | (1/5)^12 |
| 1 | 7 | 14 | 1.63840000000e-10 | (1/5)^14 |
| 1 | 8 | 16 | 6.55360000000e-12 | (1/5)^16 |
| 1 | 9 | 18 | 2.62144000000e-13 | (1/5)^18 |
| 1 | 10 | 20 | 1.04857600000e-14 | (1/5)^20 |
| 1 | 11 | 22 | 4.19430400000e-16 | (1/5)^22 |
| 1 | 12 | 24 | 1.67772160000e-17 | (1/5)^24 |
| 1 | 13 | 26 | 6.71088640000e-19 | (1/5)^26 |
| 1 | 14 | 28 | 2.68435456000e-20 | (1/5)^28 |
| 1 | 15 | 30 | 1.07374182400e-21 | (1/5)^30 |
| 1 | 16 | 32 | 4.29496729600e-23 | (1/5)^32 |
| 1 | 17 | 34 | 1.71798691840e-24 | (1/5)^34 |
| 1 | 18 | 36 | 6.87194767360e-26 | (1/5)^36 |
| 1 | 19 | 38 | 2.74877906944e-27 | (1/5)^38 |
| 1 | 20 | 40 | 1.09951162778e-28 | (1/5)^40 |
| 1 | 21 | 42 | 4.39804651110e-30 | (1/5)^42 |
| 1 | 22 | 44 | 1.75921860444e-31 | (1/5)^44 |
| 1 | 23 | 46 | 7.03687441777e-33 | (1/5)^46 |
| 1 | 24 | 48 | 2.81474976711e-34 | (1/5)^48 |
| 1 | 25 | 50 | 1.12589990684e-35 | (1/5)^50 |
| 1 | 26 | 52 | 4.50359962737e-37 | (1/5)^52 |
| 1 | 27 | 54 | 1.80143985095e-38 | (1/5)^54 |
| 1 | 28 | 56 | 7.20575940379e-40 | (1/5)^56 |
| 1 | 29 | 58 | 2.88230376152e-41 | (1/5)^58 |
| 1 | 30 | 60 | 1.15292150461e-42 | (1/5)^60 |
| 1 | 31 | 62 | 4.61168601843e-44 | (1/5)^62 |
| 1 | 32 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 1 | 33 | 66 | 7.37869762948e-47 | (1/5)^66 |
| 1 | 34 | 68 | 2.95147905179e-48 | (1/5)^68 |
| 1 | 35 | 70 | 1.18059162072e-49 | (1/5)^70 |
| 1 | 36 | 72 | 4.72236648287e-51 | (1/5)^72 |
| 1 | 37 | 74 | 1.88894659315e-52 | (1/5)^74 |
| 1 | 38 | 76 | 7.55578637259e-54 | (1/5)^76 |
| 1 | 39 | 78 | 3.02231454904e-55 | (1/5)^78 |
| 1 | 40 | 80 | 1.20892581961e-56 | (1/5)^80 |
| 1 | 41 | 82 | 4.83570327846e-58 | (1/5)^82 |
| 1 | 42 | 84 | 1.93428131138e-59 | (1/5)^84 |
| 1 | 43 | 86 | 7.73712524553e-61 | (1/5)^86 |
| 1 | 44 | 88 | 3.09485009821e-62 | (1/5)^88 |
| 1 | 45 | 90 | 1.23794003929e-63 | (1/5)^90 |
| 1 | 46 | 92 | 4.95176015714e-65 | (1/5)^92 |
| 1 | 47 | 94 | 1.98070406286e-66 | (1/5)^94 |
| 1 | 48 | 96 | 7.92281625143e-68 | (1/5)^96 |
| 1 | 49 | 98 | 3.16912650057e-69 | (1/5)^98 |
| 1 | 50 | 100 | 1.26765060023e-70 | (1/5)^100 |
| 1 | 51 | 102 | 5.07060240091e-72 | (1/5)^102 |
| 1 | 52 | 104 | 2.02824096037e-73 | (1/5)^104 |
| 1 | 53 | 106 | 8.11296384146e-75 | (1/5)^106 |
| 1 | 54 | 108 | 3.24518553658e-76 | (1/5)^108 |
| 1 | 55 | 110 | 1.29807421463e-77 | (1/5)^110 |
| 1 | 56 | 112 | 5.19229685853e-79 | (1/5)^112 |
| 1 | 57 | 114 | 2.07691874341e-80 | (1/5)^114 |
| 1 | 58 | 116 | 8.30767497366e-82 | (1/5)^116 |
| 1 | 59 | 118 | 3.32306998946e-83 | (1/5)^118 |
| 1 | 60 | 120 | 1.32922799578e-84 | (1/5)^120 |
| 1 | 61 | 122 | 5.31691198314e-86 | (1/5)^122 |
| 1 | 62 | 124 | 2.12676479326e-87 | (1/5)^124 |
| 1 | 63 | 126 | 8.50705917302e-89 | (1/5)^126 |
| 1 | 64 | 128 | 3.40282366921e-90 | (1/5)^128 |
| 2 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 2 | 1 | 4 | 1.60000000000e-3 | (1/5)^4 |
| 2 | 2 | 8 | 2.56000000000e-6 | (1/5)^8 |
| 2 | 3 | 12 | 4.09600000000e-9 | (1/5)^12 |
| 2 | 4 | 16 | 6.55360000000e-12 | (1/5)^16 |
| 2 | 5 | 20 | 1.04857600000e-14 | (1/5)^20 |
| 2 | 6 | 24 | 1.67772160000e-17 | (1/5)^24 |
| 2 | 7 | 28 | 2.68435456000e-20 | (1/5)^28 |
| 2 | 8 | 32 | 4.29496729600e-23 | (1/5)^32 |
| 2 | 9 | 36 | 6.87194767360e-26 | (1/5)^36 |
| 2 | 10 | 40 | 1.09951162778e-28 | (1/5)^40 |
| 2 | 11 | 44 | 1.75921860444e-31 | (1/5)^44 |
| 2 | 12 | 48 | 2.81474976711e-34 | (1/5)^48 |
| 2 | 13 | 52 | 4.50359962737e-37 | (1/5)^52 |
| 2 | 14 | 56 | 7.20575940379e-40 | (1/5)^56 |
| 2 | 15 | 60 | 1.15292150461e-42 | (1/5)^60 |
| 2 | 16 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 2 | 17 | 68 | 2.95147905179e-48 | (1/5)^68 |
| 2 | 18 | 72 | 4.72236648287e-51 | (1/5)^72 |
| 2 | 19 | 76 | 7.55578637259e-54 | (1/5)^76 |
| 2 | 20 | 80 | 1.20892581961e-56 | (1/5)^80 |
| 2 | 21 | 84 | 1.93428131138e-59 | (1/5)^84 |
| 2 | 22 | 88 | 3.09485009821e-62 | (1/5)^88 |
| 2 | 23 | 92 | 4.95176015714e-65 | (1/5)^92 |
| 2 | 24 | 96 | 7.92281625143e-68 | (1/5)^96 |
| 2 | 25 | 100 | 1.26765060023e-70 | (1/5)^100 |
| 2 | 26 | 104 | 2.02824096037e-73 | (1/5)^104 |
| 2 | 27 | 108 | 3.24518553658e-76 | (1/5)^108 |
| 2 | 28 | 112 | 5.19229685853e-79 | (1/5)^112 |
| 2 | 29 | 116 | 8.30767497366e-82 | (1/5)^116 |
| 2 | 30 | 120 | 1.32922799578e-84 | (1/5)^120 |
| 2 | 31 | 124 | 2.12676479326e-87 | (1/5)^124 |
| 2 | 32 | 128 | 3.40282366921e-90 | (1/5)^128 |
| 2 | 33 | 132 | 5.44451787074e-93 | (1/5)^132 |
| 2 | 34 | 136 | 8.71122859318e-96 | (1/5)^136 |
| 2 | 35 | 140 | 1.39379657491e-98 | (1/5)^140 |
| 2 | 36 | 144 | 2.23007451985e-101 | (1/5)^144 |
| 2 | 37 | 148 | 3.56811923176e-104 | (1/5)^148 |
| 2 | 38 | 152 | 5.70899077082e-107 | (1/5)^152 |
| 2 | 39 | 156 | 9.13438523332e-110 | (1/5)^156 |
| 2 | 40 | 160 | 1.46150163733e-112 | (1/5)^160 |
| 2 | 41 | 164 | 2.33840261973e-115 | (1/5)^164 |
| 2 | 42 | 168 | 3.74144419157e-118 | (1/5)^168 |
| 2 | 43 | 172 | 5.98631070651e-121 | (1/5)^172 |
| 2 | 44 | 176 | 9.57809713041e-124 | (1/5)^176 |
| 2 | 45 | 180 | 1.53249554087e-126 | (1/5)^180 |
| 2 | 46 | 184 | 2.45199286539e-129 | (1/5)^184 |
| 2 | 47 | 188 | 3.92318858462e-132 | (1/5)^188 |
| 2 | 48 | 192 | 6.27710173539e-135 | (1/5)^192 |
| 2 | 49 | 196 | 1.00433627766e-137 | (1/5)^196 |
| 2 | 50 | 200 | 1.60693804426e-140 | (1/5)^200 |
| 2 | 51 | 204 | 2.57110087081e-143 | (1/5)^204 |
| 2 | 52 | 208 | 4.11376139330e-146 | (1/5)^208 |
| 2 | 53 | 212 | 6.58201822928e-149 | (1/5)^212 |
| 2 | 54 | 216 | 1.05312291669e-151 | (1/5)^216 |
| 2 | 55 | 220 | 1.68499666670e-154 | (1/5)^220 |
| 2 | 56 | 224 | 2.69599466672e-157 | (1/5)^224 |
| 2 | 57 | 228 | 4.31359146674e-160 | (1/5)^228 |
| 2 | 58 | 232 | 6.90174634679e-163 | (1/5)^232 |
| 2 | 59 | 236 | 1.10427941549e-165 | (1/5)^236 |
| 2 | 60 | 240 | 1.76684706478e-168 | (1/5)^240 |
| 2 | 61 | 244 | 2.82695530365e-171 | (1/5)^244 |
| 2 | 62 | 248 | 4.52312848583e-174 | (1/5)^248 |
| 2 | 63 | 252 | 7.23700557733e-177 | (1/5)^252 |
| 2 | 64 | 256 | 1.15792089237e-179 | (1/5)^256 |
| 3 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 3 | 1 | 8 | 2.56000000000e-6 | (1/5)^8 |
| 3 | 2 | 16 | 6.55360000000e-12 | (1/5)^16 |
| 3 | 3 | 24 | 1.67772160000e-17 | (1/5)^24 |
| 3 | 4 | 32 | 4.29496729600e-23 | (1/5)^32 |
| 3 | 5 | 40 | 1.09951162778e-28 | (1/5)^40 |
| 3 | 6 | 48 | 2.81474976711e-34 | (1/5)^48 |
| 3 | 7 | 56 | 7.20575940379e-40 | (1/5)^56 |
| 3 | 8 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 3 | 9 | 72 | 4.72236648287e-51 | (1/5)^72 |
| 3 | 10 | 80 | 1.20892581961e-56 | (1/5)^80 |
| 3 | 11 | 88 | 3.09485009821e-62 | (1/5)^88 |
| 3 | 12 | 96 | 7.92281625143e-68 | (1/5)^96 |
| 3 | 13 | 104 | 2.02824096037e-73 | (1/5)^104 |
| 3 | 14 | 112 | 5.19229685853e-79 | (1/5)^112 |
| 3 | 15 | 120 | 1.32922799578e-84 | (1/5)^120 |
| 3 | 16 | 128 | 3.40282366921e-90 | (1/5)^128 |
| 3 | 17 | 136 | 8.71122859318e-96 | (1/5)^136 |
| 3 | 18 | 144 | 2.23007451985e-101 | (1/5)^144 |
| 3 | 19 | 152 | 5.70899077082e-107 | (1/5)^152 |
| 3 | 20 | 160 | 1.46150163733e-112 | (1/5)^160 |
| 3 | 21 | 168 | 3.74144419157e-118 | (1/5)^168 |
| 3 | 22 | 176 | 9.57809713041e-124 | (1/5)^176 |
| 3 | 23 | 184 | 2.45199286539e-129 | (1/5)^184 |
| 3 | 24 | 192 | 6.27710173539e-135 | (1/5)^192 |
| 3 | 25 | 200 | 1.60693804426e-140 | (1/5)^200 |
| 3 | 26 | 208 | 4.11376139330e-146 | (1/5)^208 |
| 3 | 27 | 216 | 1.05312291669e-151 | (1/5)^216 |
| 3 | 28 | 224 | 2.69599466672e-157 | (1/5)^224 |
| 3 | 29 | 232 | 6.90174634679e-163 | (1/5)^232 |
| 3 | 30 | 240 | 1.76684706478e-168 | (1/5)^240 |
| 3 | 31 | 248 | 4.52312848583e-174 | (1/5)^248 |
| 3 | 32 | 256 | 1.15792089237e-179 | (1/5)^256 |
| 3 | 33 | 264 | 2.96427748448e-185 | (1/5)^264 |
| 3 | 34 | 272 | 7.58855036026e-191 | (1/5)^272 |
| 3 | 35 | 280 | 1.94266889223e-196 | (1/5)^280 |
| 3 | 36 | 288 | 4.97323236410e-202 | (1/5)^288 |
| 3 | 37 | 296 | 1.27314748521e-207 | (1/5)^296 |
| 3 | 38 | 304 | 3.25925756214e-213 | (1/5)^304 |
| 3 | 39 | 312 | 8.34369935907e-219 | (1/5)^312 |
| 3 | 40 | 320 | 2.13598703592e-224 | (1/5)^320 |
| 3 | 41 | 328 | 5.46812681196e-230 | (1/5)^328 |
| 3 | 42 | 336 | 1.39984046386e-235 | (1/5)^336 |
| 3 | 43 | 344 | 3.58359158748e-241 | (1/5)^344 |
| 3 | 44 | 352 | 9.17399446396e-247 | (1/5)^352 |
| 3 | 45 | 360 | 2.34854258277e-252 | (1/5)^360 |
| 3 | 46 | 368 | 6.01226901190e-258 | (1/5)^368 |
| 3 | 47 | 376 | 1.53914086705e-263 | (1/5)^376 |
| 3 | 48 | 384 | 3.94020061964e-269 | (1/5)^384 |
| 3 | 49 | 392 | 1.00869135863e-274 | (1/5)^392 |
| 3 | 50 | 400 | 2.58224987809e-280 | (1/5)^400 |
| 3 | 51 | 408 | 6.61055968790e-286 | (1/5)^408 |
| 3 | 52 | 416 | 1.69230328010e-291 | (1/5)^416 |
| 3 | 53 | 424 | 4.33229639706e-297 | (1/5)^424 |
| 3 | 54 | 432 | 1.10906787765e-302 | (1/5)^432 |
| 3 | 55 | 440 | 2.83921376678e-308 | (1/5)^440 |
| 3 | 56 | 448 | 7.26838724296e-314 | (1/5)^448 |
| 3 | 57 | 456 | 1.86070713420e-319 | (1/5)^456 |
| 3 | 58 | 464 | 4.76341026354e-325 | (1/5)^464 |
| 3 | 59 | 472 | 1.21943302747e-330 | (1/5)^472 |
| 3 | 60 | 480 | 3.12174855032e-336 | (1/5)^480 |
| 3 | 61 | 488 | 7.99167628881e-342 | (1/5)^488 |
| 3 | 62 | 496 | 2.04586912994e-347 | (1/5)^496 |
| 3 | 63 | 504 | 5.23742497263e-353 | (1/5)^504 |
| 3 | 64 | 512 | 1.34078079299e-358 | (1/5)^512 |
| 4 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 4 | 1 | 16 | 6.55360000000e-12 | (1/5)^16 |
| 4 | 2 | 32 | 4.29496729600e-23 | (1/5)^32 |
| 4 | 3 | 48 | 2.81474976711e-34 | (1/5)^48 |
| 4 | 4 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 4 | 5 | 80 | 1.20892581961e-56 | (1/5)^80 |
| 4 | 6 | 96 | 7.92281625143e-68 | (1/5)^96 |
| 4 | 7 | 112 | 5.19229685853e-79 | (1/5)^112 |
| 4 | 8 | 128 | 3.40282366921e-90 | (1/5)^128 |
| 4 | 9 | 144 | 2.23007451985e-101 | (1/5)^144 |
| 4 | 10 | 160 | 1.46150163733e-112 | (1/5)^160 |
| 4 | 11 | 176 | 9.57809713041e-124 | (1/5)^176 |
| 4 | 12 | 192 | 6.27710173539e-135 | (1/5)^192 |
| 4 | 13 | 208 | 4.11376139330e-146 | (1/5)^208 |
| 4 | 14 | 224 | 2.69599466672e-157 | (1/5)^224 |
| 4 | 15 | 240 | 1.76684706478e-168 | (1/5)^240 |
| 4 | 16 | 256 | 1.15792089237e-179 | (1/5)^256 |
| 4 | 17 | 272 | 7.58855036026e-191 | (1/5)^272 |
| 4 | 18 | 288 | 4.97323236410e-202 | (1/5)^288 |
| 4 | 19 | 304 | 3.25925756214e-213 | (1/5)^304 |
| 4 | 20 | 320 | 2.13598703592e-224 | (1/5)^320 |
| 4 | 21 | 336 | 1.39984046386e-235 | (1/5)^336 |
| 4 | 22 | 352 | 9.17399446396e-247 | (1/5)^352 |
| 4 | 23 | 368 | 6.01226901190e-258 | (1/5)^368 |
| 4 | 24 | 384 | 3.94020061964e-269 | (1/5)^384 |
| 4 | 25 | 400 | 2.58224987809e-280 | (1/5)^400 |
| 4 | 26 | 416 | 1.69230328010e-291 | (1/5)^416 |
| 4 | 27 | 432 | 1.10906787765e-302 | (1/5)^432 |
| 4 | 28 | 448 | 7.26838724296e-314 | (1/5)^448 |
| 4 | 29 | 464 | 4.76341026354e-325 | (1/5)^464 |
| 4 | 30 | 480 | 3.12174855032e-336 | (1/5)^480 |
| 4 | 31 | 496 | 2.04586912994e-347 | (1/5)^496 |
| 4 | 32 | 512 | 1.34078079299e-358 | (1/5)^512 |
| 4 | 33 | 528 | 8.78694100497e-370 | (1/5)^528 |
| 4 | 34 | 544 | 5.75860965702e-381 | (1/5)^544 |
| 4 | 35 | 560 | 3.77396242482e-392 | (1/5)^560 |
| 4 | 36 | 576 | 2.47330401473e-403 | (1/5)^576 |
| 4 | 37 | 592 | 1.62090451909e-414 | (1/5)^592 |
| 4 | 38 | 608 | 1.06227598563e-425 | (1/5)^608 |
| 4 | 39 | 624 | 6.96173189945e-437 | (1/5)^624 |
| 4 | 40 | 640 | 4.56244061762e-448 | (1/5)^640 |
| 4 | 41 | 656 | 2.99004108316e-459 | (1/5)^656 |
| 4 | 42 | 672 | 1.95955332426e-470 | (1/5)^672 |
| 4 | 43 | 688 | 1.28421286659e-481 | (1/5)^688 |
| 4 | 44 | 704 | 8.41621744248e-493 | (1/5)^704 |
| 4 | 45 | 720 | 5.51565226310e-504 | (1/5)^720 |
| 4 | 46 | 736 | 3.61473786715e-515 | (1/5)^736 |
| 4 | 47 | 752 | 2.36895460861e-526 | (1/5)^752 |
| 4 | 48 | 768 | 1.55251809230e-537 | (1/5)^768 |
| 4 | 49 | 784 | 1.01745825697e-548 | (1/5)^784 |
| 4 | 50 | 800 | 6.66801443288e-560 | (1/5)^800 |
| 4 | 51 | 816 | 4.36994993873e-571 | (1/5)^816 |
| 4 | 52 | 832 | 2.86389039185e-582 | (1/5)^832 |
| 4 | 53 | 848 | 1.87687920720e-593 | (1/5)^848 |
| 4 | 54 | 864 | 1.23003155723e-604 | (1/5)^864 |
| 4 | 55 | 880 | 8.06113481347e-616 | (1/5)^880 |
| 4 | 56 | 896 | 5.28294531136e-627 | (1/5)^896 |
| 4 | 57 | 912 | 3.46223103925e-638 | (1/5)^912 |
| 4 | 58 | 928 | 2.26900773388e-649 | (1/5)^928 |
| 4 | 59 | 944 | 1.48701690848e-660 | (1/5)^944 |
| 4 | 60 | 960 | 9.74531401140e-672 | (1/5)^960 |
| 4 | 61 | 976 | 6.38668899051e-683 | (1/5)^976 |
| 4 | 62 | 992 | 4.18558049682e-694 | (1/5)^992 |
| 4 | 63 | 1008 | 2.74306203440e-705 | (1/5)^1008 |
| 4 | 64 | 1024 | 1.79769313486e-716 | (1/5)^1024 |
| 5 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 5 | 1 | 32 | 4.29496729600e-23 | (1/5)^32 |
| 5 | 2 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 5 | 3 | 96 | 7.92281625143e-68 | (1/5)^96 |
| 5 | 4 | 128 | 3.40282366921e-90 | (1/5)^128 |
| 5 | 5 | 160 | 1.46150163733e-112 | (1/5)^160 |
| 5 | 6 | 192 | 6.27710173539e-135 | (1/5)^192 |
| 5 | 7 | 224 | 2.69599466672e-157 | (1/5)^224 |
| 5 | 8 | 256 | 1.15792089237e-179 | (1/5)^256 |
| 5 | 9 | 288 | 4.97323236410e-202 | (1/5)^288 |
| 5 | 10 | 320 | 2.13598703592e-224 | (1/5)^320 |
| 5 | 11 | 352 | 9.17399446396e-247 | (1/5)^352 |
| 5 | 12 | 384 | 3.94020061964e-269 | (1/5)^384 |
| 5 | 13 | 416 | 1.69230328010e-291 | (1/5)^416 |
| 5 | 14 | 448 | 7.26838724296e-314 | (1/5)^448 |
| 5 | 15 | 480 | 3.12174855032e-336 | (1/5)^480 |
| 5 | 16 | 512 | 1.34078079299e-358 | (1/5)^512 |
| 5 | 17 | 544 | 5.75860965702e-381 | (1/5)^544 |
| 5 | 18 | 576 | 2.47330401473e-403 | (1/5)^576 |
| 5 | 19 | 608 | 1.06227598563e-425 | (1/5)^608 |
| 5 | 20 | 640 | 4.56244061762e-448 | (1/5)^640 |
| 5 | 21 | 672 | 1.95955332426e-470 | (1/5)^672 |
| 5 | 22 | 704 | 8.41621744248e-493 | (1/5)^704 |
| 5 | 23 | 736 | 3.61473786715e-515 | (1/5)^736 |
| 5 | 24 | 768 | 1.55251809230e-537 | (1/5)^768 |
| 5 | 25 | 800 | 6.66801443288e-560 | (1/5)^800 |
| 5 | 26 | 832 | 2.86389039185e-582 | (1/5)^832 |
| 5 | 27 | 864 | 1.23003155723e-604 | (1/5)^864 |
| 5 | 28 | 896 | 5.28294531136e-627 | (1/5)^896 |
| 5 | 29 | 928 | 2.26900773388e-649 | (1/5)^928 |
| 5 | 30 | 960 | 9.74531401140e-672 | (1/5)^960 |
| 5 | 31 | 992 | 4.18558049682e-694 | (1/5)^992 |
| 5 | 32 | 1024 | 1.79769313486e-716 | (1/5)^1024 |
| 5 | 33 | 1056 | 7.72103322248e-739 | (1/5)^1056 |
| 5 | 34 | 1088 | 3.31615851819e-761 | (1/5)^1088 |
| 5 | 35 | 1120 | 1.42427923840e-783 | (1/5)^1120 |
| 5 | 36 | 1152 | 6.11723274928e-806 | (1/5)^1152 |
| 5 | 37 | 1184 | 2.62733146002e-828 | (1/5)^1184 |
| 5 | 38 | 1216 | 1.12843026965e-850 | (1/5)^1216 |
| 5 | 39 | 1248 | 4.84657110398e-873 | (1/5)^1248 |
| 5 | 40 | 1280 | 2.08158643893e-895 | (1/5)^1280 |
| 5 | 41 | 1312 | 8.94034567901e-918 | (1/5)^1312 |
| 5 | 42 | 1344 | 3.83984923063e-940 | (1/5)^1344 |
| 5 | 43 | 1376 | 1.64920268671e-962 | (1/5)^1376 |
| 5 | 44 | 1408 | 7.08327160391e-985 | (1/5)^1408 |
| 5 | 45 | 1440 | 3.04224198875e-1007 | (1/5)^1440 |
| 5 | 46 | 1472 | 1.30663298482e-1029 | (1/5)^1472 |
| 5 | 47 | 1504 | 5.61194593767e-1052 | (1/5)^1504 |
| 5 | 48 | 1536 | 2.41031242692e-1074 | (1/5)^1536 |
| 5 | 49 | 1568 | 1.03522130468e-1096 | (1/5)^1568 |
| 5 | 50 | 1600 | 4.44624164771e-1119 | (1/5)^1600 |
| 5 | 51 | 1632 | 1.90964624670e-1141 | (1/5)^1632 |
| 5 | 52 | 1664 | 8.20186817652e-1164 | (1/5)^1664 |
| 5 | 53 | 1696 | 3.52267555842e-1186 | (1/5)^1696 |
| 5 | 54 | 1728 | 1.51297763179e-1208 | (1/5)^1728 |
| 5 | 55 | 1760 | 6.49818944810e-1231 | (1/5)^1760 |
| 5 | 56 | 1792 | 2.79095111628e-1253 | (1/5)^1792 |
| 5 | 57 | 1824 | 1.19870437692e-1275 | (1/5)^1824 |
| 5 | 58 | 1856 | 5.14839609642e-1298 | (1/5)^1856 |
| 5 | 59 | 1888 | 2.21121928610e-1320 | (1/5)^1888 |
| 5 | 60 | 1920 | 9.49711451808e-1343 | (1/5)^1920 |
| 5 | 61 | 1952 | 4.07897962615e-1365 | (1/5)^1952 |
| 5 | 62 | 1984 | 1.75190840954e-1387 | (1/5)^1984 |
| 5 | 63 | 2016 | 7.52438932455e-1410 | (1/5)^2016 |
| 5 | 64 | 2048 | 3.23170060713e-1432 | (1/5)^2048 |
| 6 | 0 | 0 | 1.00000000000e+0 | (1/5)^0 |
| 6 | 1 | 64 | 1.84467440737e-45 | (1/5)^64 |
| 6 | 2 | 128 | 3.40282366921e-90 | (1/5)^128 |
| 6 | 3 | 192 | 6.27710173539e-135 | (1/5)^192 |
| 6 | 4 | 256 | 1.15792089237e-179 | (1/5)^256 |
| 6 | 5 | 320 | 2.13598703592e-224 | (1/5)^320 |
| 6 | 6 | 384 | 3.94020061964e-269 | (1/5)^384 |
| 6 | 7 | 448 | 7.26838724296e-314 | (1/5)^448 |
| 6 | 8 | 512 | 1.34078079299e-358 | (1/5)^512 |
| 6 | 9 | 576 | 2.47330401473e-403 | (1/5)^576 |
| 6 | 10 | 640 | 4.56244061762e-448 | (1/5)^640 |
| 6 | 11 | 704 | 8.41621744248e-493 | (1/5)^704 |
| 6 | 12 | 768 | 1.55251809230e-537 | (1/5)^768 |
| 6 | 13 | 832 | 2.86389039185e-582 | (1/5)^832 |
| 6 | 14 | 896 | 5.28294531136e-627 | (1/5)^896 |
| 6 | 15 | 960 | 9.74531401140e-672 | (1/5)^960 |
| 6 | 16 | 1024 | 1.79769313486e-716 | (1/5)^1024 |
| 6 | 17 | 1088 | 3.31615851819e-761 | (1/5)^1088 |
| 6 | 18 | 1152 | 6.11723274928e-806 | (1/5)^1152 |
| 6 | 19 | 1216 | 1.12843026965e-850 | (1/5)^1216 |
| 6 | 20 | 1280 | 2.08158643893e-895 | (1/5)^1280 |
| 6 | 21 | 1344 | 3.83984923063e-940 | (1/5)^1344 |
| 6 | 22 | 1408 | 7.08327160391e-985 | (1/5)^1408 |
| 6 | 23 | 1472 | 1.30663298482e-1029 | (1/5)^1472 |
| 6 | 24 | 1536 | 2.41031242692e-1074 | (1/5)^1536 |
| 6 | 25 | 1600 | 4.44624164771e-1119 | (1/5)^1600 |
| 6 | 26 | 1664 | 8.20186817652e-1164 | (1/5)^1664 |
| 6 | 27 | 1728 | 1.51297763179e-1208 | (1/5)^1728 |
| 6 | 28 | 1792 | 2.79095111628e-1253 | (1/5)^1792 |
| 6 | 29 | 1856 | 5.14839609642e-1298 | (1/5)^1856 |
| 6 | 30 | 1920 | 9.49711451808e-1343 | (1/5)^1920 |
| 6 | 31 | 1984 | 1.75190840954e-1387 | (1/5)^1984 |
| 6 | 32 | 2048 | 3.23170060713e-1432 | (1/5)^2048 |
| 6 | 33 | 2112 | 5.96143540226e-1477 | (1/5)^2112 |
| 6 | 34 | 2176 | 1.09969073177e-1521 | (1/5)^2176 |
| 6 | 35 | 2240 | 2.02857134893e-1566 | (1/5)^2240 |
| 6 | 36 | 2304 | 3.74205365089e-1611 | (1/5)^2304 |
| 6 | 37 | 2368 | 6.90287060081e-1656 | (1/5)^2368 |
| 6 | 38 | 2432 | 1.27335487347e-1700 | (1/5)^2432 |
| 6 | 39 | 2496 | 2.34892514659e-1745 | (1/5)^2496 |
| 6 | 40 | 2560 | 4.33300210275e-1790 | (1/5)^2560 |
| 6 | 41 | 2624 | 7.99297808603e-1835 | (1/5)^2624 |
| 6 | 42 | 2688 | 1.47444421140e-1879 | (1/5)^2688 |
| 6 | 43 | 2752 | 2.71986950186e-1924 | (1/5)^2752 |
| 6 | 44 | 2816 | 5.01727366147e-1969 | (1/5)^2816 |
| 6 | 45 | 2880 | 9.25523631809e-2014 | (1/5)^2880 |
| 6 | 46 | 2944 | 1.70728975702e-2058 | (1/5)^2944 |
| 6 | 47 | 3008 | 3.14939372073e-2103 | (1/5)^3008 |
| 6 | 48 | 3072 | 5.80960599537e-2148 | (1/5)^3072 |
| 6 | 49 | 3136 | 1.07168314966e-2192 | (1/5)^3136 |
| 6 | 50 | 3200 | 1.97690647898e-2237 | (1/5)^3200 |
| 6 | 51 | 3264 | 3.64674878754e-2282 | (1/5)^3264 |
| 6 | 52 | 3328 | 6.72706415850e-2327 | (1/5)^3328 |
| 6 | 53 | 3392 | 1.24092430899e-2371 | (1/5)^3392 |
| 6 | 54 | 3456 | 2.28910131428e-2416 | (1/5)^3456 |
| 6 | 55 | 3520 | 4.22264661033e-2461 | (1/5)^3520 |
| 6 | 56 | 3584 | 7.78940813346e-2506 | (1/5)^3584 |
| 6 | 57 | 3648 | 1.43689218324e-2550 | (1/5)^3648 |
| 6 | 58 | 3712 | 2.65059823657e-2595 | (1/5)^3712 |
| 6 | 59 | 3776 | 4.88949073122e-2640 | (1/5)^3776 |
| 6 | 60 | 3840 | 9.01951841695e-2685 | (1/5)^3840 |
| 6 | 61 | 3904 | 1.66380747906e-2729 | (1/5)^3904 |
| 6 | 62 | 3968 | 3.06918307541e-2774 | (1/5)^3968 |
| 6 | 63 | 4032 | 5.66164347074e-2819 | (1/5)^4032 |
| 6 | 64 | 4096 | 1.04438888141e-2863 | (1/5)^4096 |
Complete spatial correlations for q₀=1/2: all seven steps and 65 retained-site distances. Scientific values are rounded to 12 significant digits; the final column is exact. No positive tail is replaced by zero.
| Step n | r in retained sites | Physical R/a=2ⁿr | Cₙ(r)≈ | Exact Cₙ(r) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 0 | 1 | 1 | 5.00000000000e-1 | (1/2)^1 |
| 0 | 2 | 2 | 2.50000000000e-1 | (1/2)^2 |
| 0 | 3 | 3 | 1.25000000000e-1 | (1/2)^3 |
| 0 | 4 | 4 | 6.25000000000e-2 | (1/2)^4 |
| 0 | 5 | 5 | 3.12500000000e-2 | (1/2)^5 |
| 0 | 6 | 6 | 1.56250000000e-2 | (1/2)^6 |
| 0 | 7 | 7 | 7.81250000000e-3 | (1/2)^7 |
| 0 | 8 | 8 | 3.90625000000e-3 | (1/2)^8 |
| 0 | 9 | 9 | 1.95312500000e-3 | (1/2)^9 |
| 0 | 10 | 10 | 9.76562500000e-4 | (1/2)^10 |
| 0 | 11 | 11 | 4.88281250000e-4 | (1/2)^11 |
| 0 | 12 | 12 | 2.44140625000e-4 | (1/2)^12 |
| 0 | 13 | 13 | 1.22070312500e-4 | (1/2)^13 |
| 0 | 14 | 14 | 6.10351562500e-5 | (1/2)^14 |
| 0 | 15 | 15 | 3.05175781250e-5 | (1/2)^15 |
| 0 | 16 | 16 | 1.52587890625e-5 | (1/2)^16 |
| 0 | 17 | 17 | 7.62939453125e-6 | (1/2)^17 |
| 0 | 18 | 18 | 3.81469726562e-6 | (1/2)^18 |
| 0 | 19 | 19 | 1.90734863281e-6 | (1/2)^19 |
| 0 | 20 | 20 | 9.53674316406e-7 | (1/2)^20 |
| 0 | 21 | 21 | 4.76837158203e-7 | (1/2)^21 |
| 0 | 22 | 22 | 2.38418579102e-7 | (1/2)^22 |
| 0 | 23 | 23 | 1.19209289551e-7 | (1/2)^23 |
| 0 | 24 | 24 | 5.96046447754e-8 | (1/2)^24 |
| 0 | 25 | 25 | 2.98023223877e-8 | (1/2)^25 |
| 0 | 26 | 26 | 1.49011611938e-8 | (1/2)^26 |
| 0 | 27 | 27 | 7.45058059692e-9 | (1/2)^27 |
| 0 | 28 | 28 | 3.72529029846e-9 | (1/2)^28 |
| 0 | 29 | 29 | 1.86264514923e-9 | (1/2)^29 |
| 0 | 30 | 30 | 9.31322574615e-10 | (1/2)^30 |
| 0 | 31 | 31 | 4.65661287308e-10 | (1/2)^31 |
| 0 | 32 | 32 | 2.32830643654e-10 | (1/2)^32 |
| 0 | 33 | 33 | 1.16415321827e-10 | (1/2)^33 |
| 0 | 34 | 34 | 5.82076609135e-11 | (1/2)^34 |
| 0 | 35 | 35 | 2.91038304567e-11 | (1/2)^35 |
| 0 | 36 | 36 | 1.45519152284e-11 | (1/2)^36 |
| 0 | 37 | 37 | 7.27595761418e-12 | (1/2)^37 |
| 0 | 38 | 38 | 3.63797880709e-12 | (1/2)^38 |
| 0 | 39 | 39 | 1.81898940355e-12 | (1/2)^39 |
| 0 | 40 | 40 | 9.09494701773e-13 | (1/2)^40 |
| 0 | 41 | 41 | 4.54747350886e-13 | (1/2)^41 |
| 0 | 42 | 42 | 2.27373675443e-13 | (1/2)^42 |
| 0 | 43 | 43 | 1.13686837722e-13 | (1/2)^43 |
| 0 | 44 | 44 | 5.68434188608e-14 | (1/2)^44 |
| 0 | 45 | 45 | 2.84217094304e-14 | (1/2)^45 |
| 0 | 46 | 46 | 1.42108547152e-14 | (1/2)^46 |
| 0 | 47 | 47 | 7.10542735760e-15 | (1/2)^47 |
| 0 | 48 | 48 | 3.55271367880e-15 | (1/2)^48 |
| 0 | 49 | 49 | 1.77635683940e-15 | (1/2)^49 |
| 0 | 50 | 50 | 8.88178419700e-16 | (1/2)^50 |
| 0 | 51 | 51 | 4.44089209850e-16 | (1/2)^51 |
| 0 | 52 | 52 | 2.22044604925e-16 | (1/2)^52 |
| 0 | 53 | 53 | 1.11022302463e-16 | (1/2)^53 |
| 0 | 54 | 54 | 5.55111512313e-17 | (1/2)^54 |
| 0 | 55 | 55 | 2.77555756156e-17 | (1/2)^55 |
| 0 | 56 | 56 | 1.38777878078e-17 | (1/2)^56 |
| 0 | 57 | 57 | 6.93889390391e-18 | (1/2)^57 |
| 0 | 58 | 58 | 3.46944695195e-18 | (1/2)^58 |
| 0 | 59 | 59 | 1.73472347598e-18 | (1/2)^59 |
| 0 | 60 | 60 | 8.67361737988e-19 | (1/2)^60 |
| 0 | 61 | 61 | 4.33680868994e-19 | (1/2)^61 |
| 0 | 62 | 62 | 2.16840434497e-19 | (1/2)^62 |
| 0 | 63 | 63 | 1.08420217249e-19 | (1/2)^63 |
| 0 | 64 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 1 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 1 | 1 | 2 | 2.50000000000e-1 | (1/2)^2 |
| 1 | 2 | 4 | 6.25000000000e-2 | (1/2)^4 |
| 1 | 3 | 6 | 1.56250000000e-2 | (1/2)^6 |
| 1 | 4 | 8 | 3.90625000000e-3 | (1/2)^8 |
| 1 | 5 | 10 | 9.76562500000e-4 | (1/2)^10 |
| 1 | 6 | 12 | 2.44140625000e-4 | (1/2)^12 |
| 1 | 7 | 14 | 6.10351562500e-5 | (1/2)^14 |
| 1 | 8 | 16 | 1.52587890625e-5 | (1/2)^16 |
| 1 | 9 | 18 | 3.81469726562e-6 | (1/2)^18 |
| 1 | 10 | 20 | 9.53674316406e-7 | (1/2)^20 |
| 1 | 11 | 22 | 2.38418579102e-7 | (1/2)^22 |
| 1 | 12 | 24 | 5.96046447754e-8 | (1/2)^24 |
| 1 | 13 | 26 | 1.49011611938e-8 | (1/2)^26 |
| 1 | 14 | 28 | 3.72529029846e-9 | (1/2)^28 |
| 1 | 15 | 30 | 9.31322574615e-10 | (1/2)^30 |
| 1 | 16 | 32 | 2.32830643654e-10 | (1/2)^32 |
| 1 | 17 | 34 | 5.82076609135e-11 | (1/2)^34 |
| 1 | 18 | 36 | 1.45519152284e-11 | (1/2)^36 |
| 1 | 19 | 38 | 3.63797880709e-12 | (1/2)^38 |
| 1 | 20 | 40 | 9.09494701773e-13 | (1/2)^40 |
| 1 | 21 | 42 | 2.27373675443e-13 | (1/2)^42 |
| 1 | 22 | 44 | 5.68434188608e-14 | (1/2)^44 |
| 1 | 23 | 46 | 1.42108547152e-14 | (1/2)^46 |
| 1 | 24 | 48 | 3.55271367880e-15 | (1/2)^48 |
| 1 | 25 | 50 | 8.88178419700e-16 | (1/2)^50 |
| 1 | 26 | 52 | 2.22044604925e-16 | (1/2)^52 |
| 1 | 27 | 54 | 5.55111512313e-17 | (1/2)^54 |
| 1 | 28 | 56 | 1.38777878078e-17 | (1/2)^56 |
| 1 | 29 | 58 | 3.46944695195e-18 | (1/2)^58 |
| 1 | 30 | 60 | 8.67361737988e-19 | (1/2)^60 |
| 1 | 31 | 62 | 2.16840434497e-19 | (1/2)^62 |
| 1 | 32 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 1 | 33 | 66 | 1.35525271561e-20 | (1/2)^66 |
| 1 | 34 | 68 | 3.38813178902e-21 | (1/2)^68 |
| 1 | 35 | 70 | 8.47032947254e-22 | (1/2)^70 |
| 1 | 36 | 72 | 2.11758236814e-22 | (1/2)^72 |
| 1 | 37 | 74 | 5.29395592034e-23 | (1/2)^74 |
| 1 | 38 | 76 | 1.32348898008e-23 | (1/2)^76 |
| 1 | 39 | 78 | 3.30872245021e-24 | (1/2)^78 |
| 1 | 40 | 80 | 8.27180612553e-25 | (1/2)^80 |
| 1 | 41 | 82 | 2.06795153138e-25 | (1/2)^82 |
| 1 | 42 | 84 | 5.16987882846e-26 | (1/2)^84 |
| 1 | 43 | 86 | 1.29246970711e-26 | (1/2)^86 |
| 1 | 44 | 88 | 3.23117426779e-27 | (1/2)^88 |
| 1 | 45 | 90 | 8.07793566946e-28 | (1/2)^90 |
| 1 | 46 | 92 | 2.01948391737e-28 | (1/2)^92 |
| 1 | 47 | 94 | 5.04870979341e-29 | (1/2)^94 |
| 1 | 48 | 96 | 1.26217744835e-29 | (1/2)^96 |
| 1 | 49 | 98 | 3.15544362088e-30 | (1/2)^98 |
| 1 | 50 | 100 | 7.88860905221e-31 | (1/2)^100 |
| 1 | 51 | 102 | 1.97215226305e-31 | (1/2)^102 |
| 1 | 52 | 104 | 4.93038065763e-32 | (1/2)^104 |
| 1 | 53 | 106 | 1.23259516441e-32 | (1/2)^106 |
| 1 | 54 | 108 | 3.08148791102e-33 | (1/2)^108 |
| 1 | 55 | 110 | 7.70371977755e-34 | (1/2)^110 |
| 1 | 56 | 112 | 1.92592994439e-34 | (1/2)^112 |
| 1 | 57 | 114 | 4.81482486097e-35 | (1/2)^114 |
| 1 | 58 | 116 | 1.20370621524e-35 | (1/2)^116 |
| 1 | 59 | 118 | 3.00926553811e-36 | (1/2)^118 |
| 1 | 60 | 120 | 7.52316384526e-37 | (1/2)^120 |
| 1 | 61 | 122 | 1.88079096132e-37 | (1/2)^122 |
| 1 | 62 | 124 | 4.70197740329e-38 | (1/2)^124 |
| 1 | 63 | 126 | 1.17549435082e-38 | (1/2)^126 |
| 1 | 64 | 128 | 2.93873587706e-39 | (1/2)^128 |
| 2 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 2 | 1 | 4 | 6.25000000000e-2 | (1/2)^4 |
| 2 | 2 | 8 | 3.90625000000e-3 | (1/2)^8 |
| 2 | 3 | 12 | 2.44140625000e-4 | (1/2)^12 |
| 2 | 4 | 16 | 1.52587890625e-5 | (1/2)^16 |
| 2 | 5 | 20 | 9.53674316406e-7 | (1/2)^20 |
| 2 | 6 | 24 | 5.96046447754e-8 | (1/2)^24 |
| 2 | 7 | 28 | 3.72529029846e-9 | (1/2)^28 |
| 2 | 8 | 32 | 2.32830643654e-10 | (1/2)^32 |
| 2 | 9 | 36 | 1.45519152284e-11 | (1/2)^36 |
| 2 | 10 | 40 | 9.09494701773e-13 | (1/2)^40 |
| 2 | 11 | 44 | 5.68434188608e-14 | (1/2)^44 |
| 2 | 12 | 48 | 3.55271367880e-15 | (1/2)^48 |
| 2 | 13 | 52 | 2.22044604925e-16 | (1/2)^52 |
| 2 | 14 | 56 | 1.38777878078e-17 | (1/2)^56 |
| 2 | 15 | 60 | 8.67361737988e-19 | (1/2)^60 |
| 2 | 16 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 2 | 17 | 68 | 3.38813178902e-21 | (1/2)^68 |
| 2 | 18 | 72 | 2.11758236814e-22 | (1/2)^72 |
| 2 | 19 | 76 | 1.32348898008e-23 | (1/2)^76 |
| 2 | 20 | 80 | 8.27180612553e-25 | (1/2)^80 |
| 2 | 21 | 84 | 5.16987882846e-26 | (1/2)^84 |
| 2 | 22 | 88 | 3.23117426779e-27 | (1/2)^88 |
| 2 | 23 | 92 | 2.01948391737e-28 | (1/2)^92 |
| 2 | 24 | 96 | 1.26217744835e-29 | (1/2)^96 |
| 2 | 25 | 100 | 7.88860905221e-31 | (1/2)^100 |
| 2 | 26 | 104 | 4.93038065763e-32 | (1/2)^104 |
| 2 | 27 | 108 | 3.08148791102e-33 | (1/2)^108 |
| 2 | 28 | 112 | 1.92592994439e-34 | (1/2)^112 |
| 2 | 29 | 116 | 1.20370621524e-35 | (1/2)^116 |
| 2 | 30 | 120 | 7.52316384526e-37 | (1/2)^120 |
| 2 | 31 | 124 | 4.70197740329e-38 | (1/2)^124 |
| 2 | 32 | 128 | 2.93873587706e-39 | (1/2)^128 |
| 2 | 33 | 132 | 1.83670992316e-40 | (1/2)^132 |
| 2 | 34 | 136 | 1.14794370197e-41 | (1/2)^136 |
| 2 | 35 | 140 | 7.17464813734e-43 | (1/2)^140 |
| 2 | 36 | 144 | 4.48415508584e-44 | (1/2)^144 |
| 2 | 37 | 148 | 2.80259692865e-45 | (1/2)^148 |
| 2 | 38 | 152 | 1.75162308041e-46 | (1/2)^152 |
| 2 | 39 | 156 | 1.09476442525e-47 | (1/2)^156 |
| 2 | 40 | 160 | 6.84227765784e-49 | (1/2)^160 |
| 2 | 41 | 164 | 4.27642353615e-50 | (1/2)^164 |
| 2 | 42 | 168 | 2.67276471009e-51 | (1/2)^168 |
| 2 | 43 | 172 | 1.67047794381e-52 | (1/2)^172 |
| 2 | 44 | 176 | 1.04404871488e-53 | (1/2)^176 |
| 2 | 45 | 180 | 6.52530446800e-55 | (1/2)^180 |
| 2 | 46 | 184 | 4.07831529250e-56 | (1/2)^184 |
| 2 | 47 | 188 | 2.54894705781e-57 | (1/2)^188 |
| 2 | 48 | 192 | 1.59309191113e-58 | (1/2)^192 |
| 2 | 49 | 196 | 9.95682444458e-60 | (1/2)^196 |
| 2 | 50 | 200 | 6.22301527786e-61 | (1/2)^200 |
| 2 | 51 | 204 | 3.88938454866e-62 | (1/2)^204 |
| 2 | 52 | 208 | 2.43086534291e-63 | (1/2)^208 |
| 2 | 53 | 212 | 1.51929083932e-64 | (1/2)^212 |
| 2 | 54 | 216 | 9.49556774576e-66 | (1/2)^216 |
| 2 | 55 | 220 | 5.93472984110e-67 | (1/2)^220 |
| 2 | 56 | 224 | 3.70920615069e-68 | (1/2)^224 |
| 2 | 57 | 228 | 2.31825384418e-69 | (1/2)^228 |
| 2 | 58 | 232 | 1.44890865261e-70 | (1/2)^232 |
| 2 | 59 | 236 | 9.05567907883e-72 | (1/2)^236 |
| 2 | 60 | 240 | 5.65979942427e-73 | (1/2)^240 |
| 2 | 61 | 244 | 3.53737464017e-74 | (1/2)^244 |
| 2 | 62 | 248 | 2.21085915010e-75 | (1/2)^248 |
| 2 | 63 | 252 | 1.38178696882e-76 | (1/2)^252 |
| 2 | 64 | 256 | 8.63616855509e-78 | (1/2)^256 |
| 3 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 3 | 1 | 8 | 3.90625000000e-3 | (1/2)^8 |
| 3 | 2 | 16 | 1.52587890625e-5 | (1/2)^16 |
| 3 | 3 | 24 | 5.96046447754e-8 | (1/2)^24 |
| 3 | 4 | 32 | 2.32830643654e-10 | (1/2)^32 |
| 3 | 5 | 40 | 9.09494701773e-13 | (1/2)^40 |
| 3 | 6 | 48 | 3.55271367880e-15 | (1/2)^48 |
| 3 | 7 | 56 | 1.38777878078e-17 | (1/2)^56 |
| 3 | 8 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 3 | 9 | 72 | 2.11758236814e-22 | (1/2)^72 |
| 3 | 10 | 80 | 8.27180612553e-25 | (1/2)^80 |
| 3 | 11 | 88 | 3.23117426779e-27 | (1/2)^88 |
| 3 | 12 | 96 | 1.26217744835e-29 | (1/2)^96 |
| 3 | 13 | 104 | 4.93038065763e-32 | (1/2)^104 |
| 3 | 14 | 112 | 1.92592994439e-34 | (1/2)^112 |
| 3 | 15 | 120 | 7.52316384526e-37 | (1/2)^120 |
| 3 | 16 | 128 | 2.93873587706e-39 | (1/2)^128 |
| 3 | 17 | 136 | 1.14794370197e-41 | (1/2)^136 |
| 3 | 18 | 144 | 4.48415508584e-44 | (1/2)^144 |
| 3 | 19 | 152 | 1.75162308041e-46 | (1/2)^152 |
| 3 | 20 | 160 | 6.84227765784e-49 | (1/2)^160 |
| 3 | 21 | 168 | 2.67276471009e-51 | (1/2)^168 |
| 3 | 22 | 176 | 1.04404871488e-53 | (1/2)^176 |
| 3 | 23 | 184 | 4.07831529250e-56 | (1/2)^184 |
| 3 | 24 | 192 | 1.59309191113e-58 | (1/2)^192 |
| 3 | 25 | 200 | 6.22301527786e-61 | (1/2)^200 |
| 3 | 26 | 208 | 2.43086534291e-63 | (1/2)^208 |
| 3 | 27 | 216 | 9.49556774576e-66 | (1/2)^216 |
| 3 | 28 | 224 | 3.70920615069e-68 | (1/2)^224 |
| 3 | 29 | 232 | 1.44890865261e-70 | (1/2)^232 |
| 3 | 30 | 240 | 5.65979942427e-73 | (1/2)^240 |
| 3 | 31 | 248 | 2.21085915010e-75 | (1/2)^248 |
| 3 | 32 | 256 | 8.63616855509e-78 | (1/2)^256 |
| 3 | 33 | 264 | 3.37350334183e-80 | (1/2)^264 |
| 3 | 34 | 272 | 1.31777474290e-82 | (1/2)^272 |
| 3 | 35 | 280 | 5.14755758947e-85 | (1/2)^280 |
| 3 | 36 | 288 | 2.01076468339e-87 | (1/2)^288 |
| 3 | 37 | 296 | 7.85454954448e-90 | (1/2)^296 |
| 3 | 38 | 304 | 3.06818341581e-92 | (1/2)^304 |
| 3 | 39 | 312 | 1.19850914680e-94 | (1/2)^312 |
| 3 | 40 | 320 | 4.68167635469e-97 | (1/2)^320 |
| 3 | 41 | 328 | 1.82877982605e-99 | (1/2)^328 |
| 3 | 42 | 336 | 7.14367119551e-102 | (1/2)^336 |
| 3 | 43 | 344 | 2.79049656075e-104 | (1/2)^344 |
| 3 | 44 | 352 | 1.09003771904e-106 | (1/2)^352 |
| 3 | 45 | 360 | 4.25795984001e-109 | (1/2)^360 |
| 3 | 46 | 368 | 1.66326556250e-111 | (1/2)^368 |
| 3 | 47 | 376 | 6.49713110353e-114 | (1/2)^376 |
| 3 | 48 | 384 | 2.53794183732e-116 | (1/2)^384 |
| 3 | 49 | 392 | 9.91383530201e-119 | (1/2)^392 |
| 3 | 50 | 400 | 3.87259191485e-121 | (1/2)^400 |
| 3 | 51 | 408 | 1.51273121674e-123 | (1/2)^408 |
| 3 | 52 | 416 | 5.90910631538e-126 | (1/2)^416 |
| 3 | 53 | 424 | 2.30824465445e-128 | (1/2)^424 |
| 3 | 54 | 432 | 9.01658068143e-131 | (1/2)^432 |
| 3 | 55 | 440 | 3.52210182868e-133 | (1/2)^440 |
| 3 | 56 | 448 | 1.37582102683e-135 | (1/2)^448 |
| 3 | 57 | 456 | 5.37430088605e-138 | (1/2)^456 |
| 3 | 58 | 464 | 2.09933628361e-140 | (1/2)^464 |
| 3 | 59 | 472 | 8.20053235787e-143 | (1/2)^472 |
| 3 | 60 | 480 | 3.20333295229e-145 | (1/2)^480 |
| 3 | 61 | 488 | 1.25130193449e-147 | (1/2)^488 |
| 3 | 62 | 496 | 4.88789818160e-150 | (1/2)^496 |
| 3 | 63 | 504 | 1.90933522719e-152 | (1/2)^504 |
| 3 | 64 | 512 | 7.45834073120e-155 | (1/2)^512 |
| 4 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 4 | 1 | 16 | 1.52587890625e-5 | (1/2)^16 |
| 4 | 2 | 32 | 2.32830643654e-10 | (1/2)^32 |
| 4 | 3 | 48 | 3.55271367880e-15 | (1/2)^48 |
| 4 | 4 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 4 | 5 | 80 | 8.27180612553e-25 | (1/2)^80 |
| 4 | 6 | 96 | 1.26217744835e-29 | (1/2)^96 |
| 4 | 7 | 112 | 1.92592994439e-34 | (1/2)^112 |
| 4 | 8 | 128 | 2.93873587706e-39 | (1/2)^128 |
| 4 | 9 | 144 | 4.48415508584e-44 | (1/2)^144 |
| 4 | 10 | 160 | 6.84227765784e-49 | (1/2)^160 |
| 4 | 11 | 176 | 1.04404871488e-53 | (1/2)^176 |
| 4 | 12 | 192 | 1.59309191113e-58 | (1/2)^192 |
| 4 | 13 | 208 | 2.43086534291e-63 | (1/2)^208 |
| 4 | 14 | 224 | 3.70920615069e-68 | (1/2)^224 |
| 4 | 15 | 240 | 5.65979942427e-73 | (1/2)^240 |
| 4 | 16 | 256 | 8.63616855509e-78 | (1/2)^256 |
| 4 | 17 | 272 | 1.31777474290e-82 | (1/2)^272 |
| 4 | 18 | 288 | 2.01076468339e-87 | (1/2)^288 |
| 4 | 19 | 304 | 3.06818341581e-92 | (1/2)^304 |
| 4 | 20 | 320 | 4.68167635469e-97 | (1/2)^320 |
| 4 | 21 | 336 | 7.14367119551e-102 | (1/2)^336 |
| 4 | 22 | 352 | 1.09003771904e-106 | (1/2)^352 |
| 4 | 23 | 368 | 1.66326556250e-111 | (1/2)^368 |
| 4 | 24 | 384 | 2.53794183732e-116 | (1/2)^384 |
| 4 | 25 | 400 | 3.87259191485e-121 | (1/2)^400 |
| 4 | 26 | 416 | 5.90910631538e-126 | (1/2)^416 |
| 4 | 27 | 432 | 9.01658068143e-131 | (1/2)^432 |
| 4 | 28 | 448 | 1.37582102683e-135 | (1/2)^448 |
| 4 | 29 | 464 | 2.09933628361e-140 | (1/2)^464 |
| 4 | 30 | 480 | 3.20333295229e-145 | (1/2)^480 |
| 4 | 31 | 496 | 4.88789818160e-150 | (1/2)^496 |
| 4 | 32 | 512 | 7.45834073120e-155 | (1/2)^512 |
| 4 | 33 | 528 | 1.13805247974e-159 | (1/2)^528 |
| 4 | 34 | 544 | 1.73653027304e-164 | (1/2)^544 |
| 4 | 35 | 560 | 2.64973491369e-169 | (1/2)^560 |
| 4 | 36 | 576 | 4.04317461195e-174 | (1/2)^576 |
| 4 | 37 | 592 | 6.16939485466e-179 | (1/2)^592 |
| 4 | 38 | 608 | 9.41374947306e-184 | (1/2)^608 |
| 4 | 39 | 624 | 1.43642417497e-188 | (1/2)^624 |
| 4 | 40 | 640 | 2.19180934901e-193 | (1/2)^640 |
| 4 | 41 | 656 | 3.34443565217e-198 | (1/2)^656 |
| 4 | 42 | 672 | 5.10320381496e-203 | (1/2)^672 |
| 4 | 43 | 688 | 7.78687105554e-208 | (1/2)^688 |
| 4 | 44 | 704 | 1.18818222893e-212 | (1/2)^704 |
| 4 | 45 | 720 | 1.81302219991e-217 | (1/2)^720 |
| 4 | 46 | 736 | 2.76645233141e-222 | (1/2)^736 |
| 4 | 47 | 752 | 4.22127125764e-227 | (1/2)^752 |
| 4 | 48 | 768 | 6.44114876960e-232 | (1/2)^768 |
| 4 | 49 | 784 | 9.82841303955e-237 | (1/2)^784 |
| 4 | 50 | 800 | 1.49969681390e-241 | (1/2)^800 |
| 4 | 51 | 816 | 2.28835573409e-246 | (1/2)^816 |
| 4 | 52 | 832 | 3.49175374465e-251 | (1/2)^832 |
| 4 | 53 | 848 | 5.32799338478e-256 | (1/2)^848 |
| 4 | 54 | 864 | 8.12987271848e-261 | (1/2)^864 |
| 4 | 55 | 880 | 1.24052012916e-265 | (1/2)^880 |
| 4 | 56 | 896 | 1.89288349787e-270 | (1/2)^896 |
| 4 | 57 | 912 | 2.88831100138e-275 | (1/2)^912 |
| 4 | 58 | 928 | 4.40721283170e-280 | (1/2)^928 |
| 4 | 59 | 944 | 6.72487309525e-285 | (1/2)^944 |
| 4 | 60 | 960 | 1.02613420032e-289 | (1/2)^960 |
| 4 | 61 | 976 | 1.56575653126e-294 | (1/2)^976 |
| 4 | 62 | 992 | 2.38915486337e-299 | (1/2)^992 |
| 4 | 63 | 1008 | 3.64556100978e-304 | (1/2)^1008 |
| 4 | 64 | 1024 | 5.56268464627e-309 | (1/2)^1024 |
| 5 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 5 | 1 | 32 | 2.32830643654e-10 | (1/2)^32 |
| 5 | 2 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 5 | 3 | 96 | 1.26217744835e-29 | (1/2)^96 |
| 5 | 4 | 128 | 2.93873587706e-39 | (1/2)^128 |
| 5 | 5 | 160 | 6.84227765784e-49 | (1/2)^160 |
| 5 | 6 | 192 | 1.59309191113e-58 | (1/2)^192 |
| 5 | 7 | 224 | 3.70920615069e-68 | (1/2)^224 |
| 5 | 8 | 256 | 8.63616855509e-78 | (1/2)^256 |
| 5 | 9 | 288 | 2.01076468339e-87 | (1/2)^288 |
| 5 | 10 | 320 | 4.68167635469e-97 | (1/2)^320 |
| 5 | 11 | 352 | 1.09003771904e-106 | (1/2)^352 |
| 5 | 12 | 384 | 2.53794183732e-116 | (1/2)^384 |
| 5 | 13 | 416 | 5.90910631538e-126 | (1/2)^416 |
| 5 | 14 | 448 | 1.37582102683e-135 | (1/2)^448 |
| 5 | 15 | 480 | 3.20333295229e-145 | (1/2)^480 |
| 5 | 16 | 512 | 7.45834073120e-155 | (1/2)^512 |
| 5 | 17 | 544 | 1.73653027304e-164 | (1/2)^544 |
| 5 | 18 | 576 | 4.04317461195e-174 | (1/2)^576 |
| 5 | 19 | 608 | 9.41374947306e-184 | (1/2)^608 |
| 5 | 20 | 640 | 2.19180934901e-193 | (1/2)^640 |
| 5 | 21 | 672 | 5.10320381496e-203 | (1/2)^672 |
| 5 | 22 | 704 | 1.18818222893e-212 | (1/2)^704 |
| 5 | 23 | 736 | 2.76645233141e-222 | (1/2)^736 |
| 5 | 24 | 768 | 6.44114876960e-232 | (1/2)^768 |
| 5 | 25 | 800 | 1.49969681390e-241 | (1/2)^800 |
| 5 | 26 | 832 | 3.49175374465e-251 | (1/2)^832 |
| 5 | 27 | 864 | 8.12987271848e-261 | (1/2)^864 |
| 5 | 28 | 896 | 1.89288349787e-270 | (1/2)^896 |
| 5 | 29 | 928 | 4.40721283170e-280 | (1/2)^928 |
| 5 | 30 | 960 | 1.02613420032e-289 | (1/2)^960 |
| 5 | 31 | 992 | 2.38915486337e-299 | (1/2)^992 |
| 5 | 32 | 1024 | 5.56268464627e-309 | (1/2)^1024 |
| 5 | 33 | 1056 | 1.29516344663e-318 | (1/2)^1056 |
| 5 | 34 | 1088 | 3.01553738917e-328 | (1/2)^1088 |
| 5 | 35 | 1120 | 7.02109511282e-338 | (1/2)^1120 |
| 5 | 36 | 1152 | 1.63472609427e-347 | (1/2)^1152 |
| 5 | 37 | 1184 | 3.80614328727e-357 | (1/2)^1184 |
| 5 | 38 | 1216 | 8.86186791415e-367 | (1/2)^1216 |
| 5 | 39 | 1248 | 2.06331441043e-376 | (1/2)^1248 |
| 5 | 40 | 1280 | 4.80402822240e-386 | (1/2)^1280 |
| 5 | 41 | 1312 | 1.11852498315e-395 | (1/2)^1312 |
| 5 | 42 | 1344 | 2.60426891770e-405 | (1/2)^1344 |
| 5 | 43 | 1376 | 6.06353608357e-415 | (1/2)^1376 |
| 5 | 44 | 1408 | 1.41177700916e-424 | (1/2)^1408 |
| 5 | 45 | 1440 | 3.28704949737e-434 | (1/2)^1440 |
| 5 | 46 | 1472 | 7.65325850196e-444 | (1/2)^1472 |
| 5 | 47 | 1504 | 1.78191310306e-453 | (1/2)^1504 |
| 5 | 48 | 1536 | 4.14883974721e-463 | (1/2)^1536 |
| 5 | 49 | 1568 | 9.65977028759e-473 | (1/2)^1568 |
| 5 | 50 | 1600 | 2.24909053361e-482 | (1/2)^1600 |
| 5 | 51 | 1632 | 5.23657196576e-492 | (1/2)^1632 |
| 5 | 52 | 1664 | 1.21923442133e-501 | (1/2)^1664 |
| 5 | 53 | 1696 | 2.83875135083e-511 | (1/2)^1696 |
| 5 | 54 | 1728 | 6.60948304186e-521 | (1/2)^1728 |
| 5 | 55 | 1760 | 1.53889019086e-530 | (1/2)^1760 |
| 5 | 56 | 1792 | 3.58300793650e-540 | (1/2)^1792 |
| 5 | 57 | 1824 | 8.34234044071e-550 | (1/2)^1824 |
| 5 | 58 | 1856 | 1.94235249439e-559 | (1/2)^1856 |
| 5 | 59 | 1888 | 4.52239181472e-569 | (1/2)^1888 |
| 5 | 60 | 1920 | 1.05295139708e-578 | (1/2)^1920 |
| 5 | 61 | 1952 | 2.45159351517e-588 | (1/2)^1952 |
| 5 | 62 | 1984 | 5.70806096116e-598 | (1/2)^1984 |
| 5 | 63 | 2016 | 1.32901150760e-607 | (1/2)^2016 |
| 5 | 64 | 2048 | 3.09434604738e-617 | (1/2)^2048 |
| 6 | 0 | 0 | 1.00000000000e+0 | (1/2)^0 |
| 6 | 1 | 64 | 5.42101086243e-20 | (1/2)^64 |
| 6 | 2 | 128 | 2.93873587706e-39 | (1/2)^128 |
| 6 | 3 | 192 | 1.59309191113e-58 | (1/2)^192 |
| 6 | 4 | 256 | 8.63616855509e-78 | (1/2)^256 |
| 6 | 5 | 320 | 4.68167635469e-97 | (1/2)^320 |
| 6 | 6 | 384 | 2.53794183732e-116 | (1/2)^384 |
| 6 | 7 | 448 | 1.37582102683e-135 | (1/2)^448 |
| 6 | 8 | 512 | 7.45834073120e-155 | (1/2)^512 |
| 6 | 9 | 576 | 4.04317461195e-174 | (1/2)^576 |
| 6 | 10 | 640 | 2.19180934901e-193 | (1/2)^640 |
| 6 | 11 | 704 | 1.18818222893e-212 | (1/2)^704 |
| 6 | 12 | 768 | 6.44114876960e-232 | (1/2)^768 |
| 6 | 13 | 832 | 3.49175374465e-251 | (1/2)^832 |
| 6 | 14 | 896 | 1.89288349787e-270 | (1/2)^896 |
| 6 | 15 | 960 | 1.02613420032e-289 | (1/2)^960 |
| 6 | 16 | 1024 | 5.56268464627e-309 | (1/2)^1024 |
| 6 | 17 | 1088 | 3.01553738917e-328 | (1/2)^1088 |
| 6 | 18 | 1152 | 1.63472609427e-347 | (1/2)^1152 |
| 6 | 19 | 1216 | 8.86186791415e-367 | (1/2)^1216 |
| 6 | 20 | 1280 | 4.80402822240e-386 | (1/2)^1280 |
| 6 | 21 | 1344 | 2.60426891770e-405 | (1/2)^1344 |
| 6 | 22 | 1408 | 1.41177700916e-424 | (1/2)^1408 |
| 6 | 23 | 1472 | 7.65325850196e-444 | (1/2)^1472 |
| 6 | 24 | 1536 | 4.14883974721e-463 | (1/2)^1536 |
| 6 | 25 | 1600 | 2.24909053361e-482 | (1/2)^1600 |
| 6 | 26 | 1664 | 1.21923442133e-501 | (1/2)^1664 |
| 6 | 27 | 1728 | 6.60948304186e-521 | (1/2)^1728 |
| 6 | 28 | 1792 | 3.58300793650e-540 | (1/2)^1792 |
| 6 | 29 | 1856 | 1.94235249439e-559 | (1/2)^1856 |
| 6 | 30 | 1920 | 1.05295139708e-578 | (1/2)^1920 |
| 6 | 31 | 1984 | 5.70806096116e-598 | (1/2)^1984 |
| 6 | 32 | 2048 | 3.09434604738e-617 | (1/2)^2048 |
| 6 | 33 | 2112 | 1.67744835350e-636 | (1/2)^2112 |
| 6 | 34 | 2176 | 9.09346574547e-656 | (1/2)^2176 |
| 6 | 35 | 2240 | 4.92957765833e-675 | (1/2)^2240 |
| 6 | 36 | 2304 | 2.67232940330e-694 | (1/2)^2304 |
| 6 | 37 | 2368 | 1.44867267233e-713 | (1/2)^2368 |
| 6 | 38 | 2432 | 7.85327029278e-733 | (1/2)^2432 |
| 6 | 39 | 2496 | 4.25726635628e-752 | (1/2)^2496 |
| 6 | 40 | 2560 | 2.30786871616e-771 | (1/2)^2560 |
| 6 | 41 | 2624 | 1.25109813794e-790 | (1/2)^2624 |
| 6 | 42 | 2688 | 6.78221659572e-810 | (1/2)^2688 |
| 6 | 43 | 2752 | 3.67664698367e-829 | (1/2)^2752 |
| 6 | 44 | 2816 | 1.99311432358e-848 | (1/2)^2816 |
| 6 | 45 | 2880 | 1.08046943982e-867 | (1/2)^2880 |
| 6 | 46 | 2944 | 5.85723656978e-887 | (1/2)^2944 |
| 6 | 47 | 3008 | 3.17521430686e-906 | (1/2)^3008 |
| 6 | 48 | 3072 | 1.72128712480e-925 | (1/2)^3072 |
| 6 | 49 | 3136 | 9.33111620091e-945 | (1/2)^3136 |
| 6 | 50 | 3200 | 5.05840822837e-964 | (1/2)^3200 |
| 6 | 51 | 3264 | 2.74216859526e-983 | (1/2)^3264 |
| 6 | 52 | 3328 | 1.48653257415e-1002 | (1/2)^3328 |
| 6 | 53 | 3392 | 8.05850923182e-1022 | (1/2)^3392 |
| 6 | 54 | 3456 | 4.36852660807e-1041 | (1/2)^3456 |
| 6 | 55 | 3520 | 2.36818301951e-1060 | (1/2)^3520 |
| 6 | 56 | 3584 | 1.28379458730e-1079 | (1/2)^3584 |
| 6 | 57 | 3648 | 6.95946440288e-1099 | (1/2)^3648 |
| 6 | 58 | 3712 | 3.77273321247e-1118 | (1/2)^3712 |
| 6 | 59 | 3776 | 2.04520277258e-1137 | (1/2)^3776 |
| 6 | 60 | 3840 | 1.10870664460e-1156 | (1/2)^3840 |
| 6 | 61 | 3904 | 6.01031076364e-1176 | (1/2)^3904 |
| 6 | 62 | 3968 | 3.25819599363e-1195 | (1/2)^3968 |
| 6 | 63 | 4032 | 1.76627158734e-1214 | (1/2)^4032 |
| 6 | 64 | 4096 | 9.57497746095e-1234 | (1/2)^4096 |
Complete spatial correlations for q₀=4/5: all seven steps and 65 retained-site distances. Scientific values are rounded to 12 significant digits; the final column is exact. No positive tail is replaced by zero.
| Step n | r in retained sites | Physical R/a=2ⁿr | Cₙ(r)≈ | Exact Cₙ(r) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 0 | 1 | 1 | 8.00000000000e-1 | (4/5)^1 |
| 0 | 2 | 2 | 6.40000000000e-1 | (4/5)^2 |
| 0 | 3 | 3 | 5.12000000000e-1 | (4/5)^3 |
| 0 | 4 | 4 | 4.09600000000e-1 | (4/5)^4 |
| 0 | 5 | 5 | 3.27680000000e-1 | (4/5)^5 |
| 0 | 6 | 6 | 2.62144000000e-1 | (4/5)^6 |
| 0 | 7 | 7 | 2.09715200000e-1 | (4/5)^7 |
| 0 | 8 | 8 | 1.67772160000e-1 | (4/5)^8 |
| 0 | 9 | 9 | 1.34217728000e-1 | (4/5)^9 |
| 0 | 10 | 10 | 1.07374182400e-1 | (4/5)^10 |
| 0 | 11 | 11 | 8.58993459200e-2 | (4/5)^11 |
| 0 | 12 | 12 | 6.87194767360e-2 | (4/5)^12 |
| 0 | 13 | 13 | 5.49755813888e-2 | (4/5)^13 |
| 0 | 14 | 14 | 4.39804651110e-2 | (4/5)^14 |
| 0 | 15 | 15 | 3.51843720888e-2 | (4/5)^15 |
| 0 | 16 | 16 | 2.81474976711e-2 | (4/5)^16 |
| 0 | 17 | 17 | 2.25179981369e-2 | (4/5)^17 |
| 0 | 18 | 18 | 1.80143985095e-2 | (4/5)^18 |
| 0 | 19 | 19 | 1.44115188076e-2 | (4/5)^19 |
| 0 | 20 | 20 | 1.15292150461e-2 | (4/5)^20 |
| 0 | 21 | 21 | 9.22337203685e-3 | (4/5)^21 |
| 0 | 22 | 22 | 7.37869762948e-3 | (4/5)^22 |
| 0 | 23 | 23 | 5.90295810359e-3 | (4/5)^23 |
| 0 | 24 | 24 | 4.72236648287e-3 | (4/5)^24 |
| 0 | 25 | 25 | 3.77789318630e-3 | (4/5)^25 |
| 0 | 26 | 26 | 3.02231454904e-3 | (4/5)^26 |
| 0 | 27 | 27 | 2.41785163923e-3 | (4/5)^27 |
| 0 | 28 | 28 | 1.93428131138e-3 | (4/5)^28 |
| 0 | 29 | 29 | 1.54742504911e-3 | (4/5)^29 |
| 0 | 30 | 30 | 1.23794003929e-3 | (4/5)^30 |
| 0 | 31 | 31 | 9.90352031428e-4 | (4/5)^31 |
| 0 | 32 | 32 | 7.92281625143e-4 | (4/5)^32 |
| 0 | 33 | 33 | 6.33825300114e-4 | (4/5)^33 |
| 0 | 34 | 34 | 5.07060240091e-4 | (4/5)^34 |
| 0 | 35 | 35 | 4.05648192073e-4 | (4/5)^35 |
| 0 | 36 | 36 | 3.24518553658e-4 | (4/5)^36 |
| 0 | 37 | 37 | 2.59614842927e-4 | (4/5)^37 |
| 0 | 38 | 38 | 2.07691874341e-4 | (4/5)^38 |
| 0 | 39 | 39 | 1.66153499473e-4 | (4/5)^39 |
| 0 | 40 | 40 | 1.32922799578e-4 | (4/5)^40 |
| 0 | 41 | 41 | 1.06338239663e-4 | (4/5)^41 |
| 0 | 42 | 42 | 8.50705917302e-5 | (4/5)^42 |
| 0 | 43 | 43 | 6.80564733842e-5 | (4/5)^43 |
| 0 | 44 | 44 | 5.44451787074e-5 | (4/5)^44 |
| 0 | 45 | 45 | 4.35561429659e-5 | (4/5)^45 |
| 0 | 46 | 46 | 3.48449143727e-5 | (4/5)^46 |
| 0 | 47 | 47 | 2.78759314982e-5 | (4/5)^47 |
| 0 | 48 | 48 | 2.23007451985e-5 | (4/5)^48 |
| 0 | 49 | 49 | 1.78405961588e-5 | (4/5)^49 |
| 0 | 50 | 50 | 1.42724769271e-5 | (4/5)^50 |
| 0 | 51 | 51 | 1.14179815416e-5 | (4/5)^51 |
| 0 | 52 | 52 | 9.13438523332e-6 | (4/5)^52 |
| 0 | 53 | 53 | 7.30750818665e-6 | (4/5)^53 |
| 0 | 54 | 54 | 5.84600654932e-6 | (4/5)^54 |
| 0 | 55 | 55 | 4.67680523946e-6 | (4/5)^55 |
| 0 | 56 | 56 | 3.74144419157e-6 | (4/5)^56 |
| 0 | 57 | 57 | 2.99315535325e-6 | (4/5)^57 |
| 0 | 58 | 58 | 2.39452428260e-6 | (4/5)^58 |
| 0 | 59 | 59 | 1.91561942608e-6 | (4/5)^59 |
| 0 | 60 | 60 | 1.53249554087e-6 | (4/5)^60 |
| 0 | 61 | 61 | 1.22599643269e-6 | (4/5)^61 |
| 0 | 62 | 62 | 9.80797146154e-7 | (4/5)^62 |
| 0 | 63 | 63 | 7.84637716923e-7 | (4/5)^63 |
| 0 | 64 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 1 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 1 | 1 | 2 | 6.40000000000e-1 | (4/5)^2 |
| 1 | 2 | 4 | 4.09600000000e-1 | (4/5)^4 |
| 1 | 3 | 6 | 2.62144000000e-1 | (4/5)^6 |
| 1 | 4 | 8 | 1.67772160000e-1 | (4/5)^8 |
| 1 | 5 | 10 | 1.07374182400e-1 | (4/5)^10 |
| 1 | 6 | 12 | 6.87194767360e-2 | (4/5)^12 |
| 1 | 7 | 14 | 4.39804651110e-2 | (4/5)^14 |
| 1 | 8 | 16 | 2.81474976711e-2 | (4/5)^16 |
| 1 | 9 | 18 | 1.80143985095e-2 | (4/5)^18 |
| 1 | 10 | 20 | 1.15292150461e-2 | (4/5)^20 |
| 1 | 11 | 22 | 7.37869762948e-3 | (4/5)^22 |
| 1 | 12 | 24 | 4.72236648287e-3 | (4/5)^24 |
| 1 | 13 | 26 | 3.02231454904e-3 | (4/5)^26 |
| 1 | 14 | 28 | 1.93428131138e-3 | (4/5)^28 |
| 1 | 15 | 30 | 1.23794003929e-3 | (4/5)^30 |
| 1 | 16 | 32 | 7.92281625143e-4 | (4/5)^32 |
| 1 | 17 | 34 | 5.07060240091e-4 | (4/5)^34 |
| 1 | 18 | 36 | 3.24518553658e-4 | (4/5)^36 |
| 1 | 19 | 38 | 2.07691874341e-4 | (4/5)^38 |
| 1 | 20 | 40 | 1.32922799578e-4 | (4/5)^40 |
| 1 | 21 | 42 | 8.50705917302e-5 | (4/5)^42 |
| 1 | 22 | 44 | 5.44451787074e-5 | (4/5)^44 |
| 1 | 23 | 46 | 3.48449143727e-5 | (4/5)^46 |
| 1 | 24 | 48 | 2.23007451985e-5 | (4/5)^48 |
| 1 | 25 | 50 | 1.42724769271e-5 | (4/5)^50 |
| 1 | 26 | 52 | 9.13438523332e-6 | (4/5)^52 |
| 1 | 27 | 54 | 5.84600654932e-6 | (4/5)^54 |
| 1 | 28 | 56 | 3.74144419157e-6 | (4/5)^56 |
| 1 | 29 | 58 | 2.39452428260e-6 | (4/5)^58 |
| 1 | 30 | 60 | 1.53249554087e-6 | (4/5)^60 |
| 1 | 31 | 62 | 9.80797146154e-7 | (4/5)^62 |
| 1 | 32 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 1 | 33 | 66 | 4.01734511065e-7 | (4/5)^66 |
| 1 | 34 | 68 | 2.57110087081e-7 | (4/5)^68 |
| 1 | 35 | 70 | 1.64550455732e-7 | (4/5)^70 |
| 1 | 36 | 72 | 1.05312291669e-7 | (4/5)^72 |
| 1 | 37 | 74 | 6.73998666679e-8 | (4/5)^74 |
| 1 | 38 | 76 | 4.31359146674e-8 | (4/5)^76 |
| 1 | 39 | 78 | 2.76069853872e-8 | (4/5)^78 |
| 1 | 40 | 80 | 1.76684706478e-8 | (4/5)^80 |
| 1 | 41 | 82 | 1.13078212146e-8 | (4/5)^82 |
| 1 | 42 | 84 | 7.23700557733e-9 | (4/5)^84 |
| 1 | 43 | 86 | 4.63168356949e-9 | (4/5)^86 |
| 1 | 44 | 88 | 2.96427748448e-9 | (4/5)^88 |
| 1 | 45 | 90 | 1.89713759006e-9 | (4/5)^90 |
| 1 | 46 | 92 | 1.21416805764e-9 | (4/5)^92 |
| 1 | 47 | 94 | 7.77067556890e-10 | (4/5)^94 |
| 1 | 48 | 96 | 4.97323236410e-10 | (4/5)^96 |
| 1 | 49 | 98 | 3.18286871302e-10 | (4/5)^98 |
| 1 | 50 | 100 | 2.03703597633e-10 | (4/5)^100 |
| 1 | 51 | 102 | 1.30370302485e-10 | (4/5)^102 |
| 1 | 52 | 104 | 8.34369935907e-11 | (4/5)^104 |
| 1 | 53 | 106 | 5.33996758980e-11 | (4/5)^106 |
| 1 | 54 | 108 | 3.41757925747e-11 | (4/5)^108 |
| 1 | 55 | 110 | 2.18725072478e-11 | (4/5)^110 |
| 1 | 56 | 112 | 1.39984046386e-11 | (4/5)^112 |
| 1 | 57 | 114 | 8.95897896871e-12 | (4/5)^114 |
| 1 | 58 | 116 | 5.73374653998e-12 | (4/5)^116 |
| 1 | 59 | 118 | 3.66959778558e-12 | (4/5)^118 |
| 1 | 60 | 120 | 2.34854258277e-12 | (4/5)^120 |
| 1 | 61 | 122 | 1.50306725298e-12 | (4/5)^122 |
| 1 | 62 | 124 | 9.61963041904e-13 | (4/5)^124 |
| 1 | 63 | 126 | 6.15656346819e-13 | (4/5)^126 |
| 1 | 64 | 128 | 3.94020061964e-13 | (4/5)^128 |
| 2 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 2 | 1 | 4 | 4.09600000000e-1 | (4/5)^4 |
| 2 | 2 | 8 | 1.67772160000e-1 | (4/5)^8 |
| 2 | 3 | 12 | 6.87194767360e-2 | (4/5)^12 |
| 2 | 4 | 16 | 2.81474976711e-2 | (4/5)^16 |
| 2 | 5 | 20 | 1.15292150461e-2 | (4/5)^20 |
| 2 | 6 | 24 | 4.72236648287e-3 | (4/5)^24 |
| 2 | 7 | 28 | 1.93428131138e-3 | (4/5)^28 |
| 2 | 8 | 32 | 7.92281625143e-4 | (4/5)^32 |
| 2 | 9 | 36 | 3.24518553658e-4 | (4/5)^36 |
| 2 | 10 | 40 | 1.32922799578e-4 | (4/5)^40 |
| 2 | 11 | 44 | 5.44451787074e-5 | (4/5)^44 |
| 2 | 12 | 48 | 2.23007451985e-5 | (4/5)^48 |
| 2 | 13 | 52 | 9.13438523332e-6 | (4/5)^52 |
| 2 | 14 | 56 | 3.74144419157e-6 | (4/5)^56 |
| 2 | 15 | 60 | 1.53249554087e-6 | (4/5)^60 |
| 2 | 16 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 2 | 17 | 68 | 2.57110087081e-7 | (4/5)^68 |
| 2 | 18 | 72 | 1.05312291669e-7 | (4/5)^72 |
| 2 | 19 | 76 | 4.31359146674e-8 | (4/5)^76 |
| 2 | 20 | 80 | 1.76684706478e-8 | (4/5)^80 |
| 2 | 21 | 84 | 7.23700557733e-9 | (4/5)^84 |
| 2 | 22 | 88 | 2.96427748448e-9 | (4/5)^88 |
| 2 | 23 | 92 | 1.21416805764e-9 | (4/5)^92 |
| 2 | 24 | 96 | 4.97323236410e-10 | (4/5)^96 |
| 2 | 25 | 100 | 2.03703597633e-10 | (4/5)^100 |
| 2 | 26 | 104 | 8.34369935907e-11 | (4/5)^104 |
| 2 | 27 | 108 | 3.41757925747e-11 | (4/5)^108 |
| 2 | 28 | 112 | 1.39984046386e-11 | (4/5)^112 |
| 2 | 29 | 116 | 5.73374653998e-12 | (4/5)^116 |
| 2 | 30 | 120 | 2.34854258277e-12 | (4/5)^120 |
| 2 | 31 | 124 | 9.61963041904e-13 | (4/5)^124 |
| 2 | 32 | 128 | 3.94020061964e-13 | (4/5)^128 |
| 2 | 33 | 132 | 1.61390617380e-13 | (4/5)^132 |
| 2 | 34 | 136 | 6.61055968790e-14 | (4/5)^136 |
| 2 | 35 | 140 | 2.70768524816e-14 | (4/5)^140 |
| 2 | 36 | 144 | 1.10906787765e-14 | (4/5)^144 |
| 2 | 37 | 148 | 4.54274202685e-15 | (4/5)^148 |
| 2 | 38 | 152 | 1.86070713420e-15 | (4/5)^152 |
| 2 | 39 | 156 | 7.62145642167e-16 | (4/5)^156 |
| 2 | 40 | 160 | 3.12174855032e-16 | (4/5)^160 |
| 2 | 41 | 164 | 1.27866820621e-16 | (4/5)^164 |
| 2 | 42 | 168 | 5.23742497263e-17 | (4/5)^168 |
| 2 | 43 | 172 | 2.14524926879e-17 | (4/5)^172 |
| 2 | 44 | 176 | 8.78694100497e-18 | (4/5)^176 |
| 2 | 45 | 180 | 3.59913103563e-18 | (4/5)^180 |
| 2 | 46 | 184 | 1.47420407220e-18 | (4/5)^184 |
| 2 | 47 | 188 | 6.03833987971e-19 | (4/5)^188 |
| 2 | 48 | 192 | 2.47330401473e-19 | (4/5)^192 |
| 2 | 49 | 196 | 1.01306532443e-19 | (4/5)^196 |
| 2 | 50 | 200 | 4.14951556888e-20 | (4/5)^200 |
| 2 | 51 | 204 | 1.69964157701e-20 | (4/5)^204 |
| 2 | 52 | 208 | 6.96173189945e-21 | (4/5)^208 |
| 2 | 53 | 212 | 2.85152538601e-21 | (4/5)^212 |
| 2 | 54 | 216 | 1.16798479811e-21 | (4/5)^216 |
| 2 | 55 | 220 | 4.78406573306e-22 | (4/5)^220 |
| 2 | 56 | 224 | 1.95955332426e-22 | (4/5)^224 |
| 2 | 57 | 228 | 8.02633041618e-23 | (4/5)^228 |
| 2 | 58 | 232 | 3.28758493847e-23 | (4/5)^232 |
| 2 | 59 | 236 | 1.34659479080e-23 | (4/5)^236 |
| 2 | 60 | 240 | 5.51565226310e-24 | (4/5)^240 |
| 2 | 61 | 244 | 2.25921116697e-24 | (4/5)^244 |
| 2 | 62 | 248 | 9.25372893990e-25 | (4/5)^248 |
| 2 | 63 | 252 | 3.79032737378e-25 | (4/5)^252 |
| 2 | 64 | 256 | 1.55251809230e-25 | (4/5)^256 |
| 3 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 3 | 1 | 8 | 1.67772160000e-1 | (4/5)^8 |
| 3 | 2 | 16 | 2.81474976711e-2 | (4/5)^16 |
| 3 | 3 | 24 | 4.72236648287e-3 | (4/5)^24 |
| 3 | 4 | 32 | 7.92281625143e-4 | (4/5)^32 |
| 3 | 5 | 40 | 1.32922799578e-4 | (4/5)^40 |
| 3 | 6 | 48 | 2.23007451985e-5 | (4/5)^48 |
| 3 | 7 | 56 | 3.74144419157e-6 | (4/5)^56 |
| 3 | 8 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 3 | 9 | 72 | 1.05312291669e-7 | (4/5)^72 |
| 3 | 10 | 80 | 1.76684706478e-8 | (4/5)^80 |
| 3 | 11 | 88 | 2.96427748448e-9 | (4/5)^88 |
| 3 | 12 | 96 | 4.97323236410e-10 | (4/5)^96 |
| 3 | 13 | 104 | 8.34369935907e-11 | (4/5)^104 |
| 3 | 14 | 112 | 1.39984046386e-11 | (4/5)^112 |
| 3 | 15 | 120 | 2.34854258277e-12 | (4/5)^120 |
| 3 | 16 | 128 | 3.94020061964e-13 | (4/5)^128 |
| 3 | 17 | 136 | 6.61055968790e-14 | (4/5)^136 |
| 3 | 18 | 144 | 1.10906787765e-14 | (4/5)^144 |
| 3 | 19 | 152 | 1.86070713420e-15 | (4/5)^152 |
| 3 | 20 | 160 | 3.12174855032e-16 | (4/5)^160 |
| 3 | 21 | 168 | 5.23742497263e-17 | (4/5)^168 |
| 3 | 22 | 176 | 8.78694100497e-18 | (4/5)^176 |
| 3 | 23 | 184 | 1.47420407220e-18 | (4/5)^184 |
| 3 | 24 | 192 | 2.47330401473e-19 | (4/5)^192 |
| 3 | 25 | 200 | 4.14951556888e-20 | (4/5)^200 |
| 3 | 26 | 208 | 6.96173189945e-21 | (4/5)^208 |
| 3 | 27 | 216 | 1.16798479811e-21 | (4/5)^216 |
| 3 | 28 | 224 | 1.95955332426e-22 | (4/5)^224 |
| 3 | 29 | 232 | 3.28758493847e-23 | (4/5)^232 |
| 3 | 30 | 240 | 5.51565226310e-24 | (4/5)^240 |
| 3 | 31 | 248 | 9.25372893990e-25 | (4/5)^248 |
| 3 | 32 | 256 | 1.55251809230e-25 | (4/5)^256 |
| 3 | 33 | 264 | 2.60469313784e-26 | (4/5)^264 |
| 3 | 34 | 272 | 4.36994993873e-27 | (4/5)^272 |
| 3 | 35 | 280 | 7.33155940313e-28 | (4/5)^280 |
| 3 | 36 | 288 | 1.23003155723e-28 | (4/5)^288 |
| 3 | 37 | 296 | 2.06365051225e-29 | (4/5)^296 |
| 3 | 38 | 304 | 3.46223103925e-30 | (4/5)^304 |
| 3 | 39 | 312 | 5.80865979874e-31 | (4/5)^312 |
| 3 | 40 | 320 | 9.74531401140e-32 | (4/5)^320 |
| 3 | 41 | 328 | 1.63499238157e-32 | (4/5)^328 |
| 3 | 42 | 336 | 2.74306203440e-33 | (4/5)^336 |
| 3 | 43 | 344 | 4.60209442525e-34 | (4/5)^344 |
| 3 | 44 | 352 | 7.72103322248e-35 | (4/5)^352 |
| 3 | 45 | 360 | 1.29537442117e-35 | (4/5)^360 |
| 3 | 46 | 368 | 2.17327764648e-36 | (4/5)^368 |
| 3 | 47 | 376 | 3.64615485030e-37 | (4/5)^376 |
| 3 | 48 | 384 | 6.11723274928e-38 | (4/5)^384 |
| 3 | 49 | 392 | 1.02630135157e-38 | (4/5)^392 |
| 3 | 50 | 400 | 1.72184794564e-39 | (4/5)^400 |
| 3 | 51 | 408 | 2.88878149031e-40 | (4/5)^408 |
| 3 | 52 | 416 | 4.84657110398e-41 | (4/5)^416 |
| 3 | 53 | 424 | 8.13119702708e-42 | (4/5)^424 |
| 3 | 54 | 432 | 1.36418848862e-42 | (4/5)^432 |
| 3 | 55 | 440 | 2.28872849383e-43 | (4/5)^440 |
| 3 | 56 | 448 | 3.83984923063e-44 | (4/5)^448 |
| 3 | 57 | 456 | 6.44219799497e-45 | (4/5)^456 |
| 3 | 58 | 464 | 1.08082147276e-45 | (4/5)^464 |
| 3 | 59 | 472 | 1.81331753060e-46 | (4/5)^472 |
| 3 | 60 | 480 | 3.04224198875e-47 | (4/5)^480 |
| 3 | 61 | 488 | 5.10403509695e-48 | (4/5)^488 |
| 3 | 62 | 496 | 8.56314992931e-49 | (4/5)^496 |
| 3 | 63 | 504 | 1.43665816004e-49 | (4/5)^504 |
| 3 | 64 | 512 | 2.41031242692e-50 | (4/5)^512 |
| 4 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 4 | 1 | 16 | 2.81474976711e-2 | (4/5)^16 |
| 4 | 2 | 32 | 7.92281625143e-4 | (4/5)^32 |
| 4 | 3 | 48 | 2.23007451985e-5 | (4/5)^48 |
| 4 | 4 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 4 | 5 | 80 | 1.76684706478e-8 | (4/5)^80 |
| 4 | 6 | 96 | 4.97323236410e-10 | (4/5)^96 |
| 4 | 7 | 112 | 1.39984046386e-11 | (4/5)^112 |
| 4 | 8 | 128 | 3.94020061964e-13 | (4/5)^128 |
| 4 | 9 | 144 | 1.10906787765e-14 | (4/5)^144 |
| 4 | 10 | 160 | 3.12174855032e-16 | (4/5)^160 |
| 4 | 11 | 176 | 8.78694100497e-18 | (4/5)^176 |
| 4 | 12 | 192 | 2.47330401473e-19 | (4/5)^192 |
| 4 | 13 | 208 | 6.96173189945e-21 | (4/5)^208 |
| 4 | 14 | 224 | 1.95955332426e-22 | (4/5)^224 |
| 4 | 15 | 240 | 5.51565226310e-24 | (4/5)^240 |
| 4 | 16 | 256 | 1.55251809230e-25 | (4/5)^256 |
| 4 | 17 | 272 | 4.36994993873e-27 | (4/5)^272 |
| 4 | 18 | 288 | 1.23003155723e-28 | (4/5)^288 |
| 4 | 19 | 304 | 3.46223103925e-30 | (4/5)^304 |
| 4 | 20 | 320 | 9.74531401140e-32 | (4/5)^320 |
| 4 | 21 | 336 | 2.74306203440e-33 | (4/5)^336 |
| 4 | 22 | 352 | 7.72103322248e-35 | (4/5)^352 |
| 4 | 23 | 368 | 2.17327764648e-36 | (4/5)^368 |
| 4 | 24 | 384 | 6.11723274928e-38 | (4/5)^384 |
| 4 | 25 | 400 | 1.72184794564e-39 | (4/5)^400 |
| 4 | 26 | 416 | 4.84657110398e-41 | (4/5)^416 |
| 4 | 27 | 432 | 1.36418848862e-42 | (4/5)^432 |
| 4 | 28 | 448 | 3.83984923063e-44 | (4/5)^448 |
| 4 | 29 | 464 | 1.08082147276e-45 | (4/5)^464 |
| 4 | 30 | 480 | 3.04224198875e-47 | (4/5)^480 |
| 4 | 31 | 496 | 8.56314992931e-49 | (4/5)^496 |
| 4 | 32 | 512 | 2.41031242692e-50 | (4/5)^512 |
| 4 | 33 | 528 | 6.78442634233e-52 | (4/5)^528 |
| 4 | 34 | 544 | 1.90964624670e-53 | (4/5)^544 |
| 4 | 35 | 560 | 5.37517632816e-55 | (4/5)^560 |
| 4 | 36 | 576 | 1.51297763179e-56 | (4/5)^576 |
| 4 | 37 | 592 | 4.25865343670e-58 | (4/5)^592 |
| 4 | 38 | 608 | 1.19870437692e-59 | (4/5)^608 |
| 4 | 39 | 624 | 3.37405286575e-61 | (4/5)^624 |
| 4 | 40 | 640 | 9.49711451808e-63 | (4/5)^640 |
| 4 | 41 | 656 | 2.67320008779e-64 | (4/5)^656 |
| 4 | 42 | 672 | 7.52438932455e-66 | (4/5)^672 |
| 4 | 43 | 688 | 2.11792730989e-67 | (4/5)^688 |
| 4 | 44 | 704 | 5.96143540226e-69 | (4/5)^704 |
| 4 | 45 | 720 | 1.67799489101e-70 | (4/5)^720 |
| 4 | 46 | 736 | 4.72313572869e-72 | (4/5)^736 |
| 4 | 47 | 752 | 1.32944451923e-73 | (4/5)^752 |
| 4 | 48 | 768 | 3.74205365089e-75 | (4/5)^768 |
| 4 | 49 | 784 | 1.05329446423e-76 | (4/5)^784 |
| 4 | 50 | 800 | 2.96476034790e-78 | (4/5)^800 |
| 4 | 51 | 816 | 8.34505849878e-80 | (4/5)^816 |
| 4 | 52 | 832 | 2.34892514659e-81 | (4/5)^832 |
| 4 | 53 | 848 | 6.61163650932e-83 | (4/5)^848 |
| 4 | 54 | 864 | 1.86101023248e-84 | (4/5)^864 |
| 4 | 55 | 880 | 5.23827811846e-86 | (4/5)^880 |
| 4 | 56 | 896 | 1.47444421140e-87 | (4/5)^896 |
| 4 | 57 | 912 | 4.15019150064e-89 | (4/5)^912 |
| 4 | 58 | 928 | 1.16817505599e-90 | (4/5)^928 |
| 4 | 59 | 944 | 3.28812046678e-92 | (4/5)^944 |
| 4 | 60 | 960 | 9.25523631809e-94 | (4/5)^960 |
| 4 | 61 | 976 | 2.60511742709e-95 | (4/5)^976 |
| 4 | 62 | 992 | 7.33275367118e-97 | (4/5)^992 |
| 4 | 63 | 1008 | 2.06398666882e-98 | (4/5)^1008 |
| 4 | 64 | 1024 | 5.80960599537e-100 | (4/5)^1024 |
| 5 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 5 | 1 | 32 | 7.92281625143e-4 | (4/5)^32 |
| 5 | 2 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 5 | 3 | 96 | 4.97323236410e-10 | (4/5)^96 |
| 5 | 4 | 128 | 3.94020061964e-13 | (4/5)^128 |
| 5 | 5 | 160 | 3.12174855032e-16 | (4/5)^160 |
| 5 | 6 | 192 | 2.47330401473e-19 | (4/5)^192 |
| 5 | 7 | 224 | 1.95955332426e-22 | (4/5)^224 |
| 5 | 8 | 256 | 1.55251809230e-25 | (4/5)^256 |
| 5 | 9 | 288 | 1.23003155723e-28 | (4/5)^288 |
| 5 | 10 | 320 | 9.74531401140e-32 | (4/5)^320 |
| 5 | 11 | 352 | 7.72103322248e-35 | (4/5)^352 |
| 5 | 12 | 384 | 6.11723274928e-38 | (4/5)^384 |
| 5 | 13 | 416 | 4.84657110398e-41 | (4/5)^416 |
| 5 | 14 | 448 | 3.83984923063e-44 | (4/5)^448 |
| 5 | 15 | 480 | 3.04224198875e-47 | (4/5)^480 |
| 5 | 16 | 512 | 2.41031242692e-50 | (4/5)^512 |
| 5 | 17 | 544 | 1.90964624670e-53 | (4/5)^544 |
| 5 | 18 | 576 | 1.51297763179e-56 | (4/5)^576 |
| 5 | 19 | 608 | 1.19870437692e-59 | (4/5)^608 |
| 5 | 20 | 640 | 9.49711451808e-63 | (4/5)^640 |
| 5 | 21 | 672 | 7.52438932455e-66 | (4/5)^672 |
| 5 | 22 | 704 | 5.96143540226e-69 | (4/5)^704 |
| 5 | 23 | 736 | 4.72313572869e-72 | (4/5)^736 |
| 5 | 24 | 768 | 3.74205365089e-75 | (4/5)^768 |
| 5 | 25 | 800 | 2.96476034790e-78 | (4/5)^800 |
| 5 | 26 | 832 | 2.34892514659e-81 | (4/5)^832 |
| 5 | 27 | 864 | 1.86101023248e-84 | (4/5)^864 |
| 5 | 28 | 896 | 1.47444421140e-87 | (4/5)^896 |
| 5 | 29 | 928 | 1.16817505599e-90 | (4/5)^928 |
| 5 | 30 | 960 | 9.25523631809e-94 | (4/5)^960 |
| 5 | 31 | 992 | 7.33275367118e-97 | (4/5)^992 |
| 5 | 32 | 1024 | 5.80960599537e-100 | (4/5)^1024 |
| 5 | 33 | 1056 | 4.60284407945e-103 | (4/5)^1056 |
| 5 | 34 | 1088 | 3.64674878754e-106 | (4/5)^1088 |
| 5 | 35 | 1120 | 2.88925205588e-109 | (4/5)^1120 |
| 5 | 36 | 1152 | 2.28910131428e-112 | (4/5)^1152 |
| 5 | 37 | 1184 | 1.81361290940e-115 | (4/5)^1184 |
| 5 | 38 | 1216 | 1.43689218324e-118 | (4/5)^1216 |
| 5 | 39 | 1248 | 1.13842327409e-121 | (4/5)^1248 |
| 5 | 40 | 1280 | 9.01951841695e-125 | (4/5)^1280 |
| 5 | 41 | 1312 | 7.14599870939e-128 | (4/5)^1312 |
| 5 | 42 | 1344 | 5.66164347074e-131 | (4/5)^1344 |
| 5 | 43 | 1376 | 4.48561608998e-134 | (4/5)^1376 |
| 5 | 44 | 1408 | 3.55387120553e-137 | (4/5)^1408 |
| 5 | 45 | 1440 | 2.81566685427e-140 | (4/5)^1440 |
| 5 | 46 | 1472 | 2.23080111116e-143 | (4/5)^1472 |
| 5 | 47 | 1504 | 1.76742272972e-146 | (4/5)^1504 |
| 5 | 48 | 1536 | 1.40029655262e-149 | (4/5)^1536 |
| 5 | 49 | 1568 | 1.10942922839e-152 | (4/5)^1568 |
| 5 | 50 | 1600 | 8.78980392048e-156 | (4/5)^1600 |
| 5 | 51 | 1632 | 6.96400013480e-159 | (4/5)^1632 |
| 5 | 52 | 1664 | 5.51744934429e-162 | (4/5)^1664 |
| 5 | 53 | 1696 | 4.37137373314e-165 | (4/5)^1696 |
| 5 | 54 | 1728 | 3.46335908540e-168 | (4/5)^1728 |
| 5 | 55 | 1760 | 2.74395576463e-171 | (4/5)^1760 |
| 5 | 56 | 1792 | 2.17398573252e-174 | (4/5)^1792 |
| 5 | 57 | 1824 | 1.72240894920e-177 | (4/5)^1824 |
| 5 | 58 | 1856 | 1.36463296143e-180 | (4/5)^1856 |
| 5 | 59 | 1888 | 1.08117362041e-183 | (4/5)^1888 |
| 5 | 60 | 1920 | 8.56593993037e-187 | (4/5)^1920 |
| 5 | 61 | 1952 | 6.78663680891e-190 | (4/5)^1952 |
| 5 | 62 | 1984 | 5.37692764021e-193 | (4/5)^1984 |
| 5 | 63 | 2016 | 4.26004096906e-196 | (4/5)^2016 |
| 5 | 64 | 2048 | 3.37515218214e-199 | (4/5)^2048 |
| 6 | 0 | 0 | 1.00000000000e+0 | (4/5)^0 |
| 6 | 1 | 64 | 6.27710173539e-7 | (4/5)^64 |
| 6 | 2 | 128 | 3.94020061964e-13 | (4/5)^128 |
| 6 | 3 | 192 | 2.47330401473e-19 | (4/5)^192 |
| 6 | 4 | 256 | 1.55251809230e-25 | (4/5)^256 |
| 6 | 5 | 320 | 9.74531401140e-32 | (4/5)^320 |
| 6 | 6 | 384 | 6.11723274928e-38 | (4/5)^384 |
| 6 | 7 | 448 | 3.83984923063e-44 | (4/5)^448 |
| 6 | 8 | 512 | 2.41031242692e-50 | (4/5)^512 |
| 6 | 9 | 576 | 1.51297763179e-56 | (4/5)^576 |
| 6 | 10 | 640 | 9.49711451808e-63 | (4/5)^640 |
| 6 | 11 | 704 | 5.96143540226e-69 | (4/5)^704 |
| 6 | 12 | 768 | 3.74205365089e-75 | (4/5)^768 |
| 6 | 13 | 832 | 2.34892514659e-81 | (4/5)^832 |
| 6 | 14 | 896 | 1.47444421140e-87 | (4/5)^896 |
| 6 | 15 | 960 | 9.25523631809e-94 | (4/5)^960 |
| 6 | 16 | 1024 | 5.80960599537e-100 | (4/5)^1024 |
| 6 | 17 | 1088 | 3.64674878754e-106 | (4/5)^1088 |
| 6 | 18 | 1152 | 2.28910131428e-112 | (4/5)^1152 |
| 6 | 19 | 1216 | 1.43689218324e-118 | (4/5)^1216 |
| 6 | 20 | 1280 | 9.01951841695e-125 | (4/5)^1280 |
| 6 | 21 | 1344 | 5.66164347074e-131 | (4/5)^1344 |
| 6 | 22 | 1408 | 3.55387120553e-137 | (4/5)^1408 |
| 6 | 23 | 1472 | 2.23080111116e-143 | (4/5)^1472 |
| 6 | 24 | 1536 | 1.40029655262e-149 | (4/5)^1536 |
| 6 | 25 | 1600 | 8.78980392048e-156 | (4/5)^1600 |
| 6 | 26 | 1664 | 5.51744934429e-162 | (4/5)^1664 |
| 6 | 27 | 1728 | 3.46335908540e-168 | (4/5)^1728 |
| 6 | 28 | 1792 | 2.17398573252e-174 | (4/5)^1792 |
| 6 | 29 | 1856 | 1.36463296143e-180 | (4/5)^1856 |
| 6 | 30 | 1920 | 8.56593993037e-187 | (4/5)^1920 |
| 6 | 31 | 1984 | 5.37692764021e-193 | (4/5)^1984 |
| 6 | 32 | 2048 | 3.37515218214e-199 | (4/5)^2048 |
| 6 | 33 | 2112 | 2.11861736197e-205 | (4/5)^2112 |
| 6 | 34 | 2176 | 1.32987767195e-211 | (4/5)^2176 |
| 6 | 35 | 2240 | 8.34777744242e-218 | (4/5)^2240 |
| 6 | 36 | 2304 | 5.23998482705e-224 | (4/5)^2304 |
| 6 | 37 | 2368 | 3.28919178513e-230 | (4/5)^2368 |
| 6 | 38 | 2432 | 2.06465914624e-236 | (4/5)^2432 |
| 6 | 39 | 2496 | 1.29600755099e-242 | (4/5)^2496 |
| 6 | 40 | 2560 | 8.13517124737e-249 | (4/5)^2560 |
| 6 | 41 | 2624 | 5.10652975545e-255 | (4/5)^2624 |
| 6 | 42 | 2688 | 3.20542067898e-261 | (4/5)^2688 |
| 6 | 43 | 2752 | 2.01207517066e-267 | (4/5)^2752 |
| 6 | 44 | 2816 | 1.26300005455e-273 | (4/5)^2816 |
| 6 | 45 | 2880 | 7.92797983421e-280 | (4/5)^2880 |
| 6 | 46 | 2944 | 4.97647359755e-286 | (4/5)^2944 |
| 6 | 47 | 3008 | 3.12378310553e-292 | (4/5)^3008 |
| 6 | 48 | 3072 | 1.96083043527e-298 | (4/5)^3072 |
| 6 | 49 | 3136 | 1.23083321280e-304 | (4/5)^3136 |
| 6 | 50 | 3200 | 7.72606529605e-311 | (4/5)^3200 |
| 6 | 51 | 3264 | 4.84972978775e-317 | (4/5)^3264 |
| 6 | 52 | 3328 | 3.04422472669e-323 | (4/5)^3328 |
| 6 | 53 | 3392 | 1.91089083148e-329 | (4/5)^3392 |
| 6 | 54 | 3456 | 1.19948561544e-335 | (4/5)^3456 |
| 6 | 55 | 3520 | 7.52929323826e-342 | (4/5)^3520 |
| 6 | 56 | 3584 | 4.72621396521e-348 | (4/5)^3584 |
| 6 | 57 | 3648 | 2.96669258828e-354 | (4/5)^3648 |
| 6 | 58 | 3712 | 1.86222311943e-360 | (4/5)^3712 |
| 6 | 59 | 3776 | 1.16893639746e-366 | (4/5)^3776 |
| 6 | 60 | 3840 | 7.33753268907e-373 | (4/5)^3840 |
| 6 | 61 | 3904 | 4.60584391760e-379 | (4/5)^3904 |
| 6 | 62 | 3968 | 2.89113508481e-385 | (4/5)^3968 |
| 6 | 63 | 4032 | 1.81479490581e-391 | (4/5)^4032 |
| 6 | 64 | 4096 | 1.13916522526e-397 | (4/5)^4096 |
Complete spatial correlations for q₀=19/20: all seven steps and 65 retained-site distances. Scientific values are rounded to 12 significant digits; the final column is exact. No positive tail is replaced by zero.
| Step n | r in retained sites | Physical R/a=2ⁿr | Cₙ(r)≈ | Exact Cₙ(r) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 0 | 1 | 1 | 9.50000000000e-1 | (19/20)^1 |
| 0 | 2 | 2 | 9.02500000000e-1 | (19/20)^2 |
| 0 | 3 | 3 | 8.57375000000e-1 | (19/20)^3 |
| 0 | 4 | 4 | 8.14506250000e-1 | (19/20)^4 |
| 0 | 5 | 5 | 7.73780937500e-1 | (19/20)^5 |
| 0 | 6 | 6 | 7.35091890625e-1 | (19/20)^6 |
| 0 | 7 | 7 | 6.98337296094e-1 | (19/20)^7 |
| 0 | 8 | 8 | 6.63420431289e-1 | (19/20)^8 |
| 0 | 9 | 9 | 6.30249409725e-1 | (19/20)^9 |
| 0 | 10 | 10 | 5.98736939238e-1 | (19/20)^10 |
| 0 | 11 | 11 | 5.68800092276e-1 | (19/20)^11 |
| 0 | 12 | 12 | 5.40360087663e-1 | (19/20)^12 |
| 0 | 13 | 13 | 5.13342083280e-1 | (19/20)^13 |
| 0 | 14 | 14 | 4.87674979116e-1 | (19/20)^14 |
| 0 | 15 | 15 | 4.63291230160e-1 | (19/20)^15 |
| 0 | 16 | 16 | 4.40126668652e-1 | (19/20)^16 |
| 0 | 17 | 17 | 4.18120335219e-1 | (19/20)^17 |
| 0 | 18 | 18 | 3.97214318458e-1 | (19/20)^18 |
| 0 | 19 | 19 | 3.77353602535e-1 | (19/20)^19 |
| 0 | 20 | 20 | 3.58485922409e-1 | (19/20)^20 |
| 0 | 21 | 21 | 3.40561626288e-1 | (19/20)^21 |
| 0 | 22 | 22 | 3.23533544974e-1 | (19/20)^22 |
| 0 | 23 | 23 | 3.07356867725e-1 | (19/20)^23 |
| 0 | 24 | 24 | 2.91989024339e-1 | (19/20)^24 |
| 0 | 25 | 25 | 2.77389573122e-1 | (19/20)^25 |
| 0 | 26 | 26 | 2.63520094466e-1 | (19/20)^26 |
| 0 | 27 | 27 | 2.50344089742e-1 | (19/20)^27 |
| 0 | 28 | 28 | 2.37826885255e-1 | (19/20)^28 |
| 0 | 29 | 29 | 2.25935540993e-1 | (19/20)^29 |
| 0 | 30 | 30 | 2.14638763943e-1 | (19/20)^30 |
| 0 | 31 | 31 | 2.03906825746e-1 | (19/20)^31 |
| 0 | 32 | 32 | 1.93711484459e-1 | (19/20)^32 |
| 0 | 33 | 33 | 1.84025910236e-1 | (19/20)^33 |
| 0 | 34 | 34 | 1.74824614724e-1 | (19/20)^34 |
| 0 | 35 | 35 | 1.66083383988e-1 | (19/20)^35 |
| 0 | 36 | 36 | 1.57779214788e-1 | (19/20)^36 |
| 0 | 37 | 37 | 1.49890254049e-1 | (19/20)^37 |
| 0 | 38 | 38 | 1.42395741346e-1 | (19/20)^38 |
| 0 | 39 | 39 | 1.35275954279e-1 | (19/20)^39 |
| 0 | 40 | 40 | 1.28512156565e-1 | (19/20)^40 |
| 0 | 41 | 41 | 1.22086548737e-1 | (19/20)^41 |
| 0 | 42 | 42 | 1.15982221300e-1 | (19/20)^42 |
| 0 | 43 | 43 | 1.10183110235e-1 | (19/20)^43 |
| 0 | 44 | 44 | 1.04673954723e-1 | (19/20)^44 |
| 0 | 45 | 45 | 9.94402569871e-2 | (19/20)^45 |
| 0 | 46 | 46 | 9.44682441377e-2 | (19/20)^46 |
| 0 | 47 | 47 | 8.97448319309e-2 | (19/20)^47 |
| 0 | 48 | 48 | 8.52575903343e-2 | (19/20)^48 |
| 0 | 49 | 49 | 8.09947108176e-2 | (19/20)^49 |
| 0 | 50 | 50 | 7.69449752767e-2 | (19/20)^50 |
| 0 | 51 | 51 | 7.30977265129e-2 | (19/20)^51 |
| 0 | 52 | 52 | 6.94428401872e-2 | (19/20)^52 |
| 0 | 53 | 53 | 6.59706981779e-2 | (19/20)^53 |
| 0 | 54 | 54 | 6.26721632690e-2 | (19/20)^54 |
| 0 | 55 | 55 | 5.95385551055e-2 | (19/20)^55 |
| 0 | 56 | 56 | 5.65616273503e-2 | (19/20)^56 |
| 0 | 57 | 57 | 5.37335459827e-2 | (19/20)^57 |
| 0 | 58 | 58 | 5.10468686836e-2 | (19/20)^58 |
| 0 | 59 | 59 | 4.84945252494e-2 | (19/20)^59 |
| 0 | 60 | 60 | 4.60697989870e-2 | (19/20)^60 |
| 0 | 61 | 61 | 4.37663090376e-2 | (19/20)^61 |
| 0 | 62 | 62 | 4.15779935857e-2 | (19/20)^62 |
| 0 | 63 | 63 | 3.94990939064e-2 | (19/20)^63 |
| 0 | 64 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 1 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 1 | 1 | 2 | 9.02500000000e-1 | (19/20)^2 |
| 1 | 2 | 4 | 8.14506250000e-1 | (19/20)^4 |
| 1 | 3 | 6 | 7.35091890625e-1 | (19/20)^6 |
| 1 | 4 | 8 | 6.63420431289e-1 | (19/20)^8 |
| 1 | 5 | 10 | 5.98736939238e-1 | (19/20)^10 |
| 1 | 6 | 12 | 5.40360087663e-1 | (19/20)^12 |
| 1 | 7 | 14 | 4.87674979116e-1 | (19/20)^14 |
| 1 | 8 | 16 | 4.40126668652e-1 | (19/20)^16 |
| 1 | 9 | 18 | 3.97214318458e-1 | (19/20)^18 |
| 1 | 10 | 20 | 3.58485922409e-1 | (19/20)^20 |
| 1 | 11 | 22 | 3.23533544974e-1 | (19/20)^22 |
| 1 | 12 | 24 | 2.91989024339e-1 | (19/20)^24 |
| 1 | 13 | 26 | 2.63520094466e-1 | (19/20)^26 |
| 1 | 14 | 28 | 2.37826885255e-1 | (19/20)^28 |
| 1 | 15 | 30 | 2.14638763943e-1 | (19/20)^30 |
| 1 | 16 | 32 | 1.93711484459e-1 | (19/20)^32 |
| 1 | 17 | 34 | 1.74824614724e-1 | (19/20)^34 |
| 1 | 18 | 36 | 1.57779214788e-1 | (19/20)^36 |
| 1 | 19 | 38 | 1.42395741346e-1 | (19/20)^38 |
| 1 | 20 | 40 | 1.28512156565e-1 | (19/20)^40 |
| 1 | 21 | 42 | 1.15982221300e-1 | (19/20)^42 |
| 1 | 22 | 44 | 1.04673954723e-1 | (19/20)^44 |
| 1 | 23 | 46 | 9.44682441377e-2 | (19/20)^46 |
| 1 | 24 | 48 | 8.52575903343e-2 | (19/20)^48 |
| 1 | 25 | 50 | 7.69449752767e-2 | (19/20)^50 |
| 1 | 26 | 52 | 6.94428401872e-2 | (19/20)^52 |
| 1 | 27 | 54 | 6.26721632690e-2 | (19/20)^54 |
| 1 | 28 | 56 | 5.65616273503e-2 | (19/20)^56 |
| 1 | 29 | 58 | 5.10468686836e-2 | (19/20)^58 |
| 1 | 30 | 60 | 4.60697989870e-2 | (19/20)^60 |
| 1 | 31 | 62 | 4.15779935857e-2 | (19/20)^62 |
| 1 | 32 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 1 | 33 | 66 | 3.38655356380e-2 | (19/20)^66 |
| 1 | 34 | 68 | 3.05636459133e-2 | (19/20)^68 |
| 1 | 35 | 70 | 2.75836904368e-2 | (19/20)^70 |
| 1 | 36 | 72 | 2.48942806192e-2 | (19/20)^72 |
| 1 | 37 | 74 | 2.24670882588e-2 | (19/20)^74 |
| 1 | 38 | 76 | 2.02765471536e-2 | (19/20)^76 |
| 1 | 39 | 78 | 1.82995838061e-2 | (19/20)^78 |
| 1 | 40 | 80 | 1.65153743850e-2 | (19/20)^80 |
| 1 | 41 | 82 | 1.49051253825e-2 | (19/20)^82 |
| 1 | 42 | 84 | 1.34518756577e-2 | (19/20)^84 |
| 1 | 43 | 86 | 1.21403177811e-2 | (19/20)^86 |
| 1 | 44 | 88 | 1.09566367974e-2 | (19/20)^88 |
| 1 | 45 | 90 | 9.88836470966e-3 | (19/20)^90 |
| 1 | 46 | 92 | 8.92424915047e-3 | (19/20)^92 |
| 1 | 47 | 94 | 8.05413485830e-3 | (19/20)^94 |
| 1 | 48 | 96 | 7.26885670961e-3 | (19/20)^96 |
| 1 | 49 | 98 | 6.56014318043e-3 | (19/20)^98 |
| 1 | 50 | 100 | 5.92052922033e-3 | (19/20)^100 |
| 1 | 51 | 102 | 5.34327762135e-3 | (19/20)^102 |
| 1 | 52 | 104 | 4.82230805327e-3 | (19/20)^104 |
| 1 | 53 | 106 | 4.35213301808e-3 | (19/20)^106 |
| 1 | 54 | 108 | 3.92780004881e-3 | (19/20)^108 |
| 1 | 55 | 110 | 3.54483954405e-3 | (19/20)^110 |
| 1 | 56 | 112 | 3.19921768851e-3 | (19/20)^112 |
| 1 | 57 | 114 | 2.88729396388e-3 | (19/20)^114 |
| 1 | 58 | 116 | 2.60578280240e-3 | (19/20)^116 |
| 1 | 59 | 118 | 2.35171897917e-3 | (19/20)^118 |
| 1 | 60 | 120 | 2.12242637870e-3 | (19/20)^120 |
| 1 | 61 | 122 | 1.91548980678e-3 | (19/20)^122 |
| 1 | 62 | 124 | 1.72872955061e-3 | (19/20)^124 |
| 1 | 63 | 126 | 1.56017841943e-3 | (19/20)^126 |
| 1 | 64 | 128 | 1.40806102354e-3 | (19/20)^128 |
| 2 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 2 | 1 | 4 | 8.14506250000e-1 | (19/20)^4 |
| 2 | 2 | 8 | 6.63420431289e-1 | (19/20)^8 |
| 2 | 3 | 12 | 5.40360087663e-1 | (19/20)^12 |
| 2 | 4 | 16 | 4.40126668652e-1 | (19/20)^16 |
| 2 | 5 | 20 | 3.58485922409e-1 | (19/20)^20 |
| 2 | 6 | 24 | 2.91989024339e-1 | (19/20)^24 |
| 2 | 7 | 28 | 2.37826885255e-1 | (19/20)^28 |
| 2 | 8 | 32 | 1.93711484459e-1 | (19/20)^32 |
| 2 | 9 | 36 | 1.57779214788e-1 | (19/20)^36 |
| 2 | 10 | 40 | 1.28512156565e-1 | (19/20)^40 |
| 2 | 11 | 44 | 1.04673954723e-1 | (19/20)^44 |
| 2 | 12 | 48 | 8.52575903343e-2 | (19/20)^48 |
| 2 | 13 | 52 | 6.94428401872e-2 | (19/20)^52 |
| 2 | 14 | 56 | 5.65616273503e-2 | (19/20)^56 |
| 2 | 15 | 60 | 4.60697989870e-2 | (19/20)^60 |
| 2 | 16 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 2 | 17 | 68 | 3.05636459133e-2 | (19/20)^68 |
| 2 | 18 | 72 | 2.48942806192e-2 | (19/20)^72 |
| 2 | 19 | 76 | 2.02765471536e-2 | (19/20)^76 |
| 2 | 20 | 80 | 1.65153743850e-2 | (19/20)^80 |
| 2 | 21 | 84 | 1.34518756577e-2 | (19/20)^84 |
| 2 | 22 | 88 | 1.09566367974e-2 | (19/20)^88 |
| 2 | 23 | 92 | 8.92424915047e-3 | (19/20)^92 |
| 2 | 24 | 96 | 7.26885670961e-3 | (19/20)^96 |
| 2 | 25 | 100 | 5.92052922033e-3 | (19/20)^100 |
| 2 | 26 | 104 | 4.82230805327e-3 | (19/20)^104 |
| 2 | 27 | 108 | 3.92780004881e-3 | (19/20)^108 |
| 2 | 28 | 112 | 3.19921768851e-3 | (19/20)^112 |
| 2 | 29 | 116 | 2.60578280240e-3 | (19/20)^116 |
| 2 | 30 | 120 | 2.12242637870e-3 | (19/20)^120 |
| 2 | 31 | 124 | 1.72872955061e-3 | (19/20)^124 |
| 2 | 32 | 128 | 1.40806102354e-3 | (19/20)^128 |
| 2 | 33 | 132 | 1.14687450405e-3 | (19/20)^132 |
| 2 | 34 | 136 | 9.34136451515e-4 | (19/20)^136 |
| 2 | 35 | 140 | 7.60859978112e-4 | (19/20)^140 |
| 2 | 36 | 144 | 6.19725207547e-4 | (19/20)^144 |
| 2 | 37 | 148 | 5.04770054830e-4 | (19/20)^148 |
| 2 | 38 | 152 | 4.11138364472e-4 | (19/20)^152 |
| 2 | 39 | 156 | 3.34874767477e-4 | (19/20)^156 |
| 2 | 40 | 160 | 2.72757591077e-4 | (19/20)^160 |
| 2 | 41 | 164 | 2.22162762667e-4 | (19/20)^164 |
| 2 | 42 | 168 | 1.80952958710e-4 | (19/20)^168 |
| 2 | 43 | 172 | 1.47387315825e-4 | (19/20)^172 |
| 2 | 44 | 176 | 1.20047889910e-4 | (19/20)^176 |
| 2 | 45 | 180 | 9.77797566312e-5 | (19/20)^180 |
| 2 | 46 | 184 | 7.96422228996e-5 | (19/20)^184 |
| 2 | 47 | 188 | 6.48690883156e-5 | (19/20)^188 |
| 2 | 48 | 192 | 5.28362778649e-5 | (19/20)^192 |
| 2 | 49 | 196 | 4.30354785477e-5 | (19/20)^196 |
| 2 | 50 | 200 | 3.50526662488e-5 | (19/20)^200 |
| 2 | 51 | 204 | 2.85506157388e-5 | (19/20)^204 |
| 2 | 52 | 208 | 2.32546549606e-5 | (19/20)^208 |
| 2 | 53 | 212 | 1.89410618070e-5 | (19/20)^212 |
| 2 | 54 | 216 | 1.54276132235e-5 | (19/20)^216 |
| 2 | 55 | 220 | 1.25658873931e-5 | (19/20)^220 |
| 2 | 56 | 224 | 1.02349938185e-5 | (19/20)^224 |
| 2 | 57 | 228 | 8.33646643385e-6 | (19/20)^228 |
| 2 | 58 | 232 | 6.79010401329e-6 | (19/20)^232 |
| 2 | 59 | 236 | 5.53058215697e-6 | (19/20)^236 |
| 2 | 60 | 240 | 4.50469373299e-6 | (19/20)^240 |
| 2 | 61 | 244 | 3.66910119986e-6 | (19/20)^244 |
| 2 | 62 | 248 | 2.98850585917e-6 | (19/20)^248 |
| 2 | 63 | 252 | 2.43415670045e-6 | (19/20)^252 |
| 2 | 64 | 256 | 1.98263584600e-6 | (19/20)^256 |
| 3 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 3 | 1 | 8 | 6.63420431289e-1 | (19/20)^8 |
| 3 | 2 | 16 | 4.40126668652e-1 | (19/20)^16 |
| 3 | 3 | 24 | 2.91989024339e-1 | (19/20)^24 |
| 3 | 4 | 32 | 1.93711484459e-1 | (19/20)^32 |
| 3 | 5 | 40 | 1.28512156565e-1 | (19/20)^40 |
| 3 | 6 | 48 | 8.52575903343e-2 | (19/20)^48 |
| 3 | 7 | 56 | 5.65616273503e-2 | (19/20)^56 |
| 3 | 8 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 3 | 9 | 72 | 2.48942806192e-2 | (19/20)^72 |
| 3 | 10 | 80 | 1.65153743850e-2 | (19/20)^80 |
| 3 | 11 | 88 | 1.09566367974e-2 | (19/20)^88 |
| 3 | 12 | 96 | 7.26885670961e-3 | (19/20)^96 |
| 3 | 13 | 104 | 4.82230805327e-3 | (19/20)^104 |
| 3 | 14 | 112 | 3.19921768851e-3 | (19/20)^112 |
| 3 | 15 | 120 | 2.12242637870e-3 | (19/20)^120 |
| 3 | 16 | 128 | 1.40806102354e-3 | (19/20)^128 |
| 3 | 17 | 136 | 9.34136451515e-4 | (19/20)^136 |
| 3 | 18 | 144 | 6.19725207547e-4 | (19/20)^144 |
| 3 | 19 | 152 | 4.11138364472e-4 | (19/20)^152 |
| 3 | 20 | 160 | 2.72757591077e-4 | (19/20)^160 |
| 3 | 21 | 168 | 1.80952958710e-4 | (19/20)^168 |
| 3 | 22 | 176 | 1.20047889910e-4 | (19/20)^176 |
| 3 | 23 | 184 | 7.96422228996e-5 | (19/20)^184 |
| 3 | 24 | 192 | 5.28362778649e-5 | (19/20)^192 |
| 3 | 25 | 200 | 3.50526662488e-5 | (19/20)^200 |
| 3 | 26 | 208 | 2.32546549606e-5 | (19/20)^208 |
| 3 | 27 | 216 | 1.54276132235e-5 | (19/20)^216 |
| 3 | 28 | 224 | 1.02349938185e-5 | (19/20)^224 |
| 3 | 29 | 232 | 6.79010401329e-6 | (19/20)^232 |
| 3 | 30 | 240 | 4.50469373299e-6 | (19/20)^240 |
| 3 | 31 | 248 | 2.98850585917e-6 | (19/20)^248 |
| 3 | 32 | 256 | 1.98263584600e-6 | (19/20)^256 |
| 3 | 33 | 264 | 1.31532112804e-6 | (19/20)^264 |
| 3 | 34 | 272 | 8.72610910049e-7 | (19/20)^272 |
| 3 | 35 | 280 | 5.78907906292e-7 | (19/20)^280 |
| 3 | 36 | 288 | 3.84059332869e-7 | (19/20)^288 |
| 3 | 37 | 296 | 2.54792808253e-7 | (19/20)^296 |
| 3 | 38 | 304 | 1.69034754740e-7 | (19/20)^304 |
| 3 | 39 | 312 | 1.12141109893e-7 | (19/20)^312 |
| 3 | 40 | 320 | 7.43967034902e-8 | (19/20)^320 |
| 3 | 41 | 328 | 4.93562931160e-8 | (19/20)^328 |
| 3 | 42 | 336 | 3.27439732658e-8 | (19/20)^336 |
| 3 | 43 | 344 | 2.17230208661e-8 | (19/20)^344 |
| 3 | 44 | 352 | 1.44114958719e-8 | (19/20)^352 |
| 3 | 45 | 360 | 9.56088080686e-9 | (19/20)^360 |
| 3 | 46 | 368 | 6.34288366839e-9 | (19/20)^368 |
| 3 | 47 | 376 | 4.20799861890e-9 | (19/20)^376 |
| 3 | 48 | 384 | 2.79167225862e-9 | (19/20)^384 |
| 3 | 49 | 392 | 1.85205241383e-9 | (19/20)^392 |
| 3 | 50 | 400 | 1.22868941115e-9 | (19/20)^400 |
| 3 | 51 | 408 | 8.15137659067e-10 | (19/20)^408 |
| 3 | 52 | 416 | 5.40778977338e-10 | (19/20)^416 |
| 3 | 53 | 424 | 3.58763822378e-10 | (19/20)^424 |
| 3 | 54 | 432 | 2.38011249773e-10 | (19/20)^432 |
| 3 | 55 | 440 | 1.57901525976e-10 | (19/20)^440 |
| 3 | 56 | 448 | 1.04755098464e-10 | (19/20)^448 |
| 3 | 57 | 456 | 6.94966726028e-11 | (19/20)^456 |
| 3 | 58 | 464 | 4.61055125113e-11 | (19/20)^464 |
| 3 | 59 | 472 | 3.05873389950e-11 | (19/20)^472 |
| 3 | 60 | 480 | 2.02922656281e-11 | (19/20)^480 |
| 3 | 61 | 488 | 1.34623036148e-11 | (19/20)^488 |
| 3 | 62 | 496 | 8.93116727028e-12 | (19/20)^496 |
| 3 | 63 | 504 | 5.92511884236e-12 | (19/20)^504 |
| 3 | 64 | 512 | 3.93084489784e-12 | (19/20)^512 |
| 4 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 4 | 1 | 16 | 4.40126668652e-1 | (19/20)^16 |
| 4 | 2 | 32 | 1.93711484459e-1 | (19/20)^32 |
| 4 | 3 | 48 | 8.52575903343e-2 | (19/20)^48 |
| 4 | 4 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 4 | 5 | 80 | 1.65153743850e-2 | (19/20)^80 |
| 4 | 6 | 96 | 7.26885670961e-3 | (19/20)^96 |
| 4 | 7 | 112 | 3.19921768851e-3 | (19/20)^112 |
| 4 | 8 | 128 | 1.40806102354e-3 | (19/20)^128 |
| 4 | 9 | 144 | 6.19725207547e-4 | (19/20)^144 |
| 4 | 10 | 160 | 2.72757591077e-4 | (19/20)^160 |
| 4 | 11 | 176 | 1.20047889910e-4 | (19/20)^176 |
| 4 | 12 | 192 | 5.28362778649e-5 | (19/20)^192 |
| 4 | 13 | 208 | 2.32546549606e-5 | (19/20)^208 |
| 4 | 14 | 224 | 1.02349938185e-5 | (19/20)^224 |
| 4 | 15 | 240 | 4.50469373299e-6 | (19/20)^240 |
| 4 | 16 | 256 | 1.98263584600e-6 | (19/20)^256 |
| 4 | 17 | 272 | 8.72610910049e-7 | (19/20)^272 |
| 4 | 18 | 288 | 3.84059332869e-7 | (19/20)^288 |
| 4 | 19 | 304 | 1.69034754740e-7 | (19/20)^304 |
| 4 | 20 | 320 | 7.43967034902e-8 | (19/20)^320 |
| 4 | 21 | 336 | 3.27439732658e-8 | (19/20)^336 |
| 4 | 22 | 352 | 1.44114958719e-8 | (19/20)^352 |
| 4 | 23 | 368 | 6.34288366839e-9 | (19/20)^368 |
| 4 | 24 | 384 | 2.79167225862e-9 | (19/20)^384 |
| 4 | 25 | 400 | 1.22868941115e-9 | (19/20)^400 |
| 4 | 26 | 416 | 5.40778977338e-10 | (19/20)^416 |
| 4 | 27 | 432 | 2.38011249773e-10 | (19/20)^432 |
| 4 | 28 | 448 | 1.04755098464e-10 | (19/20)^448 |
| 4 | 29 | 464 | 4.61055125113e-11 | (19/20)^464 |
| 4 | 30 | 480 | 2.02922656281e-11 | (19/20)^480 |
| 4 | 31 | 496 | 8.93116727028e-12 | (19/20)^496 |
| 4 | 32 | 512 | 3.93084489784e-12 | (19/20)^512 |
| 4 | 33 | 528 | 1.73006966987e-12 | (19/20)^528 |
| 4 | 34 | 544 | 7.61449800337e-13 | (19/20)^544 |
| 4 | 35 | 560 | 3.35134363968e-13 | (19/20)^560 |
| 4 | 36 | 576 | 1.47501571164e-13 | (19/20)^576 |
| 4 | 37 | 592 | 6.49193751373e-14 | (19/20)^592 |
| 4 | 38 | 608 | 2.85727483101e-14 | (19/20)^608 |
| 4 | 39 | 624 | 1.25756285280e-14 | (19/20)^624 |
| 4 | 40 | 640 | 5.53486949021e-15 | (19/20)^640 |
| 4 | 41 | 656 | 2.43604367015e-15 | (19/20)^656 |
| 4 | 42 | 672 | 1.07216778523e-15 | (19/20)^672 |
| 4 | 43 | 688 | 4.71889635550e-16 | (19/20)^688 |
| 4 | 44 | 704 | 2.07691213266e-16 | (19/20)^704 |
| 4 | 45 | 720 | 9.14104418030e-17 | (19/20)^720 |
| 4 | 46 | 736 | 4.02321732308e-17 | (19/20)^736 |
| 4 | 47 | 752 | 1.77072523767e-17 | (19/20)^752 |
| 4 | 48 | 768 | 7.79343399952e-18 | (19/20)^768 |
| 4 | 49 | 784 | 3.43009814357e-18 | (19/20)^784 |
| 4 | 50 | 800 | 1.50967766908e-18 | (19/20)^800 |
| 4 | 51 | 816 | 6.64449403229e-19 | (19/20)^816 |
| 4 | 52 | 832 | 2.92441902331e-19 | (19/20)^832 |
| 4 | 53 | 848 | 1.28711480247e-19 | (19/20)^848 |
| 4 | 54 | 864 | 5.66493550183e-20 | (19/20)^864 |
| 4 | 55 | 880 | 2.49328919055e-20 | (19/20)^880 |
| 4 | 56 | 896 | 1.09736306542e-20 | (19/20)^896 |
| 4 | 57 | 912 | 4.82978750286e-21 | (19/20)^912 |
| 4 | 58 | 928 | 2.12571828393e-21 | (19/20)^928 |
| 4 | 59 | 944 | 9.35585306797e-22 | (19/20)^944 |
| 4 | 60 | 960 | 4.11776044320e-22 | (19/20)^960 |
| 4 | 61 | 976 | 1.81233618617e-22 | (19/20)^976 |
| 4 | 62 | 992 | 7.97657488097e-23 | (19/20)^992 |
| 4 | 63 | 1008 | 3.51070332961e-23 | (19/20)^1008 |
| 4 | 64 | 1024 | 1.54515416109e-23 | (19/20)^1024 |
| 5 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 5 | 1 | 32 | 1.93711484459e-1 | (19/20)^32 |
| 5 | 2 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 5 | 3 | 96 | 7.26885670961e-3 | (19/20)^96 |
| 5 | 4 | 128 | 1.40806102354e-3 | (19/20)^128 |
| 5 | 5 | 160 | 2.72757591077e-4 | (19/20)^160 |
| 5 | 6 | 192 | 5.28362778649e-5 | (19/20)^192 |
| 5 | 7 | 224 | 1.02349938185e-5 | (19/20)^224 |
| 5 | 8 | 256 | 1.98263584600e-6 | (19/20)^256 |
| 5 | 9 | 288 | 3.84059332869e-7 | (19/20)^288 |
| 5 | 10 | 320 | 7.43967034902e-8 | (19/20)^320 |
| 5 | 11 | 352 | 1.44114958719e-8 | (19/20)^352 |
| 5 | 12 | 384 | 2.79167225862e-9 | (19/20)^384 |
| 5 | 13 | 416 | 5.40778977338e-10 | (19/20)^416 |
| 5 | 14 | 448 | 1.04755098464e-10 | (19/20)^448 |
| 5 | 15 | 480 | 2.02922656281e-11 | (19/20)^480 |
| 5 | 16 | 512 | 3.93084489784e-12 | (19/20)^512 |
| 5 | 17 | 544 | 7.61449800337e-13 | (19/20)^544 |
| 5 | 18 | 576 | 1.47501571164e-13 | (19/20)^576 |
| 5 | 19 | 608 | 2.85727483101e-14 | (19/20)^608 |
| 5 | 20 | 640 | 5.53486949021e-15 | (19/20)^640 |
| 5 | 21 | 672 | 1.07216778523e-15 | (19/20)^672 |
| 5 | 22 | 704 | 2.07691213266e-16 | (19/20)^704 |
| 5 | 23 | 736 | 4.02321732308e-17 | (19/20)^736 |
| 5 | 24 | 768 | 7.79343399952e-18 | (19/20)^768 |
| 5 | 25 | 800 | 1.50967766908e-18 | (19/20)^800 |
| 5 | 26 | 832 | 2.92441902331e-19 | (19/20)^832 |
| 5 | 27 | 864 | 5.66493550183e-20 | (19/20)^864 |
| 5 | 28 | 896 | 1.09736306542e-20 | (19/20)^896 |
| 5 | 29 | 928 | 2.12571828393e-21 | (19/20)^928 |
| 5 | 30 | 960 | 4.11776044320e-22 | (19/20)^960 |
| 5 | 31 | 992 | 7.97657488097e-23 | (19/20)^992 |
| 5 | 32 | 1024 | 1.54515416109e-23 | (19/20)^1024 |
| 5 | 33 | 1056 | 2.99314106262e-24 | (19/20)^1056 |
| 5 | 34 | 1088 | 5.79805798433e-25 | (19/20)^1088 |
| 5 | 35 | 1120 | 1.12315041912e-25 | (19/20)^1120 |
| 5 | 36 | 1152 | 2.17567134958e-26 | (19/20)^1152 |
| 5 | 37 | 1184 | 4.21452526821e-27 | (19/20)^1184 |
| 5 | 38 | 1216 | 8.16401945993e-28 | (19/20)^1216 |
| 5 | 39 | 1248 | 1.58146432873e-28 | (19/20)^1248 |
| 5 | 40 | 1280 | 3.06347802737e-29 | (19/20)^1280 |
| 5 | 41 | 1312 | 5.93430876287e-30 | (19/20)^1312 |
| 5 | 42 | 1344 | 1.14954375969e-30 | (19/20)^1344 |
| 5 | 43 | 1376 | 2.22679828140e-31 | (19/20)^1376 |
| 5 | 44 | 1408 | 4.31356400679e-32 | (19/20)^1408 |
| 5 | 45 | 1440 | 8.35586887063e-33 | (19/20)^1440 |
| 5 | 46 | 1472 | 1.61862776287e-33 | (19/20)^1472 |
| 5 | 47 | 1504 | 3.13546786731e-34 | (19/20)^1504 |
| 5 | 48 | 1536 | 6.07376135049e-35 | (19/20)^1536 |
| 5 | 49 | 1568 | 1.17655732745e-35 | (19/20)^1568 |
| 5 | 50 | 1600 | 2.27912666451e-36 | (19/20)^1600 |
| 5 | 51 | 1632 | 4.41493009451e-37 | (19/20)^1632 |
| 5 | 52 | 1664 | 8.55222662388e-38 | (19/20)^1664 |
| 5 | 53 | 1696 | 1.65666451474e-38 | (19/20)^1696 |
| 5 | 54 | 1728 | 3.20914942399e-39 | (19/20)^1728 |
| 5 | 55 | 1760 | 6.21649098771e-40 | (19/20)^1760 |
| 5 | 56 | 1792 | 1.20420569735e-40 | (19/20)^1792 |
| 5 | 57 | 1824 | 2.33268473227e-41 | (19/20)^1824 |
| 5 | 58 | 1856 | 4.51867822263e-42 | (19/20)^1856 |
| 5 | 59 | 1888 | 8.75319866295e-43 | (19/20)^1888 |
| 5 | 60 | 1920 | 1.69559510676e-43 | (19/20)^1920 |
| 5 | 61 | 1952 | 3.28456245171e-44 | (19/20)^1952 |
| 5 | 62 | 1984 | 6.36257468318e-45 | (19/20)^1984 |
| 5 | 63 | 2016 | 1.23250378686e-45 | (19/20)^2016 |
| 5 | 64 | 2048 | 2.38750138153e-46 | (19/20)^2048 |
| 6 | 0 | 0 | 1.00000000000e+0 | (19/20)^0 |
| 6 | 1 | 64 | 3.75241392111e-2 | (19/20)^64 |
| 6 | 2 | 128 | 1.40806102354e-3 | (19/20)^128 |
| 6 | 3 | 192 | 5.28362778649e-5 | (19/20)^192 |
| 6 | 4 | 256 | 1.98263584600e-6 | (19/20)^256 |
| 6 | 5 | 320 | 7.43967034902e-8 | (19/20)^320 |
| 6 | 6 | 384 | 2.79167225862e-9 | (19/20)^384 |
| 6 | 7 | 448 | 1.04755098464e-10 | (19/20)^448 |
| 6 | 8 | 512 | 3.93084489784e-12 | (19/20)^512 |
| 6 | 9 | 576 | 1.47501571164e-13 | (19/20)^576 |
| 6 | 10 | 640 | 5.53486949021e-15 | (19/20)^640 |
| 6 | 11 | 704 | 2.07691213266e-16 | (19/20)^704 |
| 6 | 12 | 768 | 7.79343399952e-18 | (19/20)^768 |
| 6 | 13 | 832 | 2.92441902331e-19 | (19/20)^832 |
| 6 | 14 | 896 | 1.09736306542e-20 | (19/20)^896 |
| 6 | 15 | 960 | 4.11776044320e-22 | (19/20)^960 |
| 6 | 16 | 1024 | 1.54515416109e-23 | (19/20)^1024 |
| 6 | 17 | 1088 | 5.79805798433e-25 | (19/20)^1088 |
| 6 | 18 | 1152 | 2.17567134958e-26 | (19/20)^1152 |
| 6 | 19 | 1216 | 8.16401945993e-28 | (19/20)^1216 |
| 6 | 20 | 1280 | 3.06347802737e-29 | (19/20)^1280 |
| 6 | 21 | 1344 | 1.14954375969e-30 | (19/20)^1344 |
| 6 | 22 | 1408 | 4.31356400679e-32 | (19/20)^1408 |
| 6 | 23 | 1472 | 1.61862776287e-33 | (19/20)^1472 |
| 6 | 24 | 1536 | 6.07376135049e-35 | (19/20)^1536 |
| 6 | 25 | 1600 | 2.27912666451e-36 | (19/20)^1600 |
| 6 | 26 | 1664 | 8.55222662388e-38 | (19/20)^1664 |
| 6 | 27 | 1728 | 3.20914942399e-39 | (19/20)^1728 |
| 6 | 28 | 1792 | 1.20420569735e-40 | (19/20)^1792 |
| 6 | 29 | 1856 | 4.51867822263e-42 | (19/20)^1856 |
| 6 | 30 | 1920 | 1.69559510676e-43 | (19/20)^1920 |
| 6 | 31 | 1984 | 6.36257468318e-45 | (19/20)^1984 |
| 6 | 32 | 2048 | 2.38750138153e-46 | (19/20)^2048 |
| 6 | 33 | 2112 | 8.95889342071e-48 | (19/20)^2112 |
| 6 | 34 | 2176 | 3.36174763896e-49 | (19/20)^2176 |
| 6 | 35 | 2240 | 1.26146686397e-50 | (19/20)^2240 |
| 6 | 36 | 2304 | 4.73354582139e-52 | (19/20)^2304 |
| 6 | 37 | 2368 | 1.77622232364e-53 | (19/20)^2368 |
| 6 | 38 | 2432 | 6.66512137421e-55 | (19/20)^2432 |
| 6 | 39 | 2496 | 2.50102942305e-56 | (19/20)^2496 |
| 6 | 40 | 2560 | 9.38489762416e-58 | (19/20)^2560 |
| 6 | 41 | 2624 | 3.52160204931e-59 | (19/20)^2624 |
| 6 | 42 | 2688 | 1.32145085545e-60 | (19/20)^2688 |
| 6 | 43 | 2752 | 4.95863058604e-62 | (19/20)^2752 |
| 6 | 44 | 2816 | 1.86068344407e-63 | (19/20)^2816 |
| 6 | 45 | 2880 | 6.98205445831e-65 | (19/20)^2880 |
| 6 | 46 | 2944 | 2.61995583473e-66 | (19/20)^2944 |
| 6 | 47 | 3008 | 9.83115874694e-68 | (19/20)^3008 |
| 6 | 48 | 3072 | 3.68905769427e-69 | (19/20)^3072 |
| 6 | 49 | 3136 | 1.38428714478e-70 | (19/20)^3136 |
| 6 | 50 | 3200 | 5.19441835287e-72 | (19/20)^3200 |
| 6 | 51 | 3264 | 1.94916077394e-73 | (19/20)^3264 |
| 6 | 52 | 3328 | 7.31405802262e-75 | (19/20)^3328 |
| 6 | 53 | 3392 | 2.74453731439e-76 | (19/20)^3392 |
| 6 | 54 | 3456 | 1.02986400255e-77 | (19/20)^3456 |
| 6 | 55 | 3520 | 3.86447602003e-79 | (19/20)^3520 |
| 6 | 56 | 3584 | 1.45011136154e-80 | (19/20)^3584 |
| 6 | 57 | 3648 | 5.44141806019e-82 | (19/20)^3648 |
| 6 | 58 | 3712 | 2.04184528796e-83 | (19/20)^3712 |
| 6 | 59 | 3776 | 7.66184868331e-85 | (19/20)^3776 |
| 6 | 60 | 3840 | 2.87504276607e-86 | (19/20)^3840 |
| 6 | 61 | 3904 | 1.07883504992e-87 | (19/20)^3904 |
| 6 | 62 | 3968 | 4.04823565990e-89 | (19/20)^3968 |
| 6 | 63 | 4032 | 1.51906558462e-90 | (19/20)^4032 |
| 6 | 64 | 4096 | 5.70016284679e-92 | (19/20)^4096 |
Complete spatial correlations for q₀=99/100: all seven steps and 65 retained-site distances. Scientific values are rounded to 12 significant digits; the final column is exact. No positive tail is replaced by zero.
| Step n | r in retained sites | Physical R/a=2ⁿr | Cₙ(r)≈ | Exact Cₙ(r) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 0 | 1 | 1 | 9.90000000000e-1 | (99/100)^1 |
| 0 | 2 | 2 | 9.80100000000e-1 | (99/100)^2 |
| 0 | 3 | 3 | 9.70299000000e-1 | (99/100)^3 |
| 0 | 4 | 4 | 9.60596010000e-1 | (99/100)^4 |
| 0 | 5 | 5 | 9.50990049900e-1 | (99/100)^5 |
| 0 | 6 | 6 | 9.41480149401e-1 | (99/100)^6 |
| 0 | 7 | 7 | 9.32065347907e-1 | (99/100)^7 |
| 0 | 8 | 8 | 9.22744694428e-1 | (99/100)^8 |
| 0 | 9 | 9 | 9.13517247484e-1 | (99/100)^9 |
| 0 | 10 | 10 | 9.04382075009e-1 | (99/100)^10 |
| 0 | 11 | 11 | 8.95338254259e-1 | (99/100)^11 |
| 0 | 12 | 12 | 8.86384871716e-1 | (99/100)^12 |
| 0 | 13 | 13 | 8.77521022999e-1 | (99/100)^13 |
| 0 | 14 | 14 | 8.68745812769e-1 | (99/100)^14 |
| 0 | 15 | 15 | 8.60058354641e-1 | (99/100)^15 |
| 0 | 16 | 16 | 8.51457771095e-1 | (99/100)^16 |
| 0 | 17 | 17 | 8.42943193384e-1 | (99/100)^17 |
| 0 | 18 | 18 | 8.34513761450e-1 | (99/100)^18 |
| 0 | 19 | 19 | 8.26168623836e-1 | (99/100)^19 |
| 0 | 20 | 20 | 8.17906937597e-1 | (99/100)^20 |
| 0 | 21 | 21 | 8.09727868221e-1 | (99/100)^21 |
| 0 | 22 | 22 | 8.01630589539e-1 | (99/100)^22 |
| 0 | 23 | 23 | 7.93614283644e-1 | (99/100)^23 |
| 0 | 24 | 24 | 7.85678140807e-1 | (99/100)^24 |
| 0 | 25 | 25 | 7.77821359399e-1 | (99/100)^25 |
| 0 | 26 | 26 | 7.70043145805e-1 | (99/100)^26 |
| 0 | 27 | 27 | 7.62342714347e-1 | (99/100)^27 |
| 0 | 28 | 28 | 7.54719287204e-1 | (99/100)^28 |
| 0 | 29 | 29 | 7.47172094332e-1 | (99/100)^29 |
| 0 | 30 | 30 | 7.39700373388e-1 | (99/100)^30 |
| 0 | 31 | 31 | 7.32303369654e-1 | (99/100)^31 |
| 0 | 32 | 32 | 7.24980335958e-1 | (99/100)^32 |
| 0 | 33 | 33 | 7.17730532598e-1 | (99/100)^33 |
| 0 | 34 | 34 | 7.10553227272e-1 | (99/100)^34 |
| 0 | 35 | 35 | 7.03447695000e-1 | (99/100)^35 |
| 0 | 36 | 36 | 6.96413218050e-1 | (99/100)^36 |
| 0 | 37 | 37 | 6.89449085869e-1 | (99/100)^37 |
| 0 | 38 | 38 | 6.82554595010e-1 | (99/100)^38 |
| 0 | 39 | 39 | 6.75729049060e-1 | (99/100)^39 |
| 0 | 40 | 40 | 6.68971758570e-1 | (99/100)^40 |
| 0 | 41 | 41 | 6.62282040984e-1 | (99/100)^41 |
| 0 | 42 | 42 | 6.55659220574e-1 | (99/100)^42 |
| 0 | 43 | 43 | 6.49102628368e-1 | (99/100)^43 |
| 0 | 44 | 44 | 6.42611602085e-1 | (99/100)^44 |
| 0 | 45 | 45 | 6.36185486064e-1 | (99/100)^45 |
| 0 | 46 | 46 | 6.29823631203e-1 | (99/100)^46 |
| 0 | 47 | 47 | 6.23525394891e-1 | (99/100)^47 |
| 0 | 48 | 48 | 6.17290140942e-1 | (99/100)^48 |
| 0 | 49 | 49 | 6.11117239533e-1 | (99/100)^49 |
| 0 | 50 | 50 | 6.05006067138e-1 | (99/100)^50 |
| 0 | 51 | 51 | 5.98956006466e-1 | (99/100)^51 |
| 0 | 52 | 52 | 5.92966446401e-1 | (99/100)^52 |
| 0 | 53 | 53 | 5.87036781937e-1 | (99/100)^53 |
| 0 | 54 | 54 | 5.81166414118e-1 | (99/100)^54 |
| 0 | 55 | 55 | 5.75354749977e-1 | (99/100)^55 |
| 0 | 56 | 56 | 5.69601202477e-1 | (99/100)^56 |
| 0 | 57 | 57 | 5.63905190452e-1 | (99/100)^57 |
| 0 | 58 | 58 | 5.58266138548e-1 | (99/100)^58 |
| 0 | 59 | 59 | 5.52683477162e-1 | (99/100)^59 |
| 0 | 60 | 60 | 5.47156642391e-1 | (99/100)^60 |
| 0 | 61 | 61 | 5.41685075967e-1 | (99/100)^61 |
| 0 | 62 | 62 | 5.36268225207e-1 | (99/100)^62 |
| 0 | 63 | 63 | 5.30905542955e-1 | (99/100)^63 |
| 0 | 64 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 1 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 1 | 1 | 2 | 9.80100000000e-1 | (99/100)^2 |
| 1 | 2 | 4 | 9.60596010000e-1 | (99/100)^4 |
| 1 | 3 | 6 | 9.41480149401e-1 | (99/100)^6 |
| 1 | 4 | 8 | 9.22744694428e-1 | (99/100)^8 |
| 1 | 5 | 10 | 9.04382075009e-1 | (99/100)^10 |
| 1 | 6 | 12 | 8.86384871716e-1 | (99/100)^12 |
| 1 | 7 | 14 | 8.68745812769e-1 | (99/100)^14 |
| 1 | 8 | 16 | 8.51457771095e-1 | (99/100)^16 |
| 1 | 9 | 18 | 8.34513761450e-1 | (99/100)^18 |
| 1 | 10 | 20 | 8.17906937597e-1 | (99/100)^20 |
| 1 | 11 | 22 | 8.01630589539e-1 | (99/100)^22 |
| 1 | 12 | 24 | 7.85678140807e-1 | (99/100)^24 |
| 1 | 13 | 26 | 7.70043145805e-1 | (99/100)^26 |
| 1 | 14 | 28 | 7.54719287204e-1 | (99/100)^28 |
| 1 | 15 | 30 | 7.39700373388e-1 | (99/100)^30 |
| 1 | 16 | 32 | 7.24980335958e-1 | (99/100)^32 |
| 1 | 17 | 34 | 7.10553227272e-1 | (99/100)^34 |
| 1 | 18 | 36 | 6.96413218050e-1 | (99/100)^36 |
| 1 | 19 | 38 | 6.82554595010e-1 | (99/100)^38 |
| 1 | 20 | 40 | 6.68971758570e-1 | (99/100)^40 |
| 1 | 21 | 42 | 6.55659220574e-1 | (99/100)^42 |
| 1 | 22 | 44 | 6.42611602085e-1 | (99/100)^44 |
| 1 | 23 | 46 | 6.29823631203e-1 | (99/100)^46 |
| 1 | 24 | 48 | 6.17290140942e-1 | (99/100)^48 |
| 1 | 25 | 50 | 6.05006067138e-1 | (99/100)^50 |
| 1 | 26 | 52 | 5.92966446401e-1 | (99/100)^52 |
| 1 | 27 | 54 | 5.81166414118e-1 | (99/100)^54 |
| 1 | 28 | 56 | 5.69601202477e-1 | (99/100)^56 |
| 1 | 29 | 58 | 5.58266138548e-1 | (99/100)^58 |
| 1 | 30 | 60 | 5.47156642391e-1 | (99/100)^60 |
| 1 | 31 | 62 | 5.36268225207e-1 | (99/100)^62 |
| 1 | 32 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 1 | 33 | 66 | 5.15137117424e-1 | (99/100)^66 |
| 1 | 34 | 68 | 5.04885888787e-1 | (99/100)^68 |
| 1 | 35 | 70 | 4.94838659600e-1 | (99/100)^70 |
| 1 | 36 | 72 | 4.84991370274e-1 | (99/100)^72 |
| 1 | 37 | 74 | 4.75340042006e-1 | (99/100)^74 |
| 1 | 38 | 76 | 4.65880775170e-1 | (99/100)^76 |
| 1 | 39 | 78 | 4.56609747744e-1 | (99/100)^78 |
| 1 | 40 | 80 | 4.47523213764e-1 | (99/100)^80 |
| 1 | 41 | 82 | 4.38617501810e-1 | (99/100)^82 |
| 1 | 42 | 84 | 4.29889013524e-1 | (99/100)^84 |
| 1 | 43 | 86 | 4.21334222155e-1 | (99/100)^86 |
| 1 | 44 | 88 | 4.12949671134e-1 | (99/100)^88 |
| 1 | 45 | 90 | 4.04731972678e-1 | (99/100)^90 |
| 1 | 46 | 92 | 3.96677806422e-1 | (99/100)^92 |
| 1 | 47 | 94 | 3.88783918074e-1 | (99/100)^94 |
| 1 | 48 | 96 | 3.81047118105e-1 | (99/100)^96 |
| 1 | 49 | 98 | 3.73464280454e-1 | (99/100)^98 |
| 1 | 50 | 100 | 3.66032341273e-1 | (99/100)^100 |
| 1 | 51 | 102 | 3.58748297682e-1 | (99/100)^102 |
| 1 | 52 | 104 | 3.51609206558e-1 | (99/100)^104 |
| 1 | 53 | 106 | 3.44612183348e-1 | (99/100)^106 |
| 1 | 54 | 108 | 3.37754400899e-1 | (99/100)^108 |
| 1 | 55 | 110 | 3.31033088321e-1 | (99/100)^110 |
| 1 | 56 | 112 | 3.24445529863e-1 | (99/100)^112 |
| 1 | 57 | 114 | 3.17989063819e-1 | (99/100)^114 |
| 1 | 58 | 116 | 3.11661081449e-1 | (99/100)^116 |
| 1 | 59 | 118 | 3.05459025928e-1 | (99/100)^118 |
| 1 | 60 | 120 | 2.99380391312e-1 | (99/100)^120 |
| 1 | 61 | 122 | 2.93422721525e-1 | (99/100)^122 |
| 1 | 62 | 124 | 2.87583609367e-1 | (99/100)^124 |
| 1 | 63 | 126 | 2.81860695540e-1 | (99/100)^126 |
| 1 | 64 | 128 | 2.76251667699e-1 | (99/100)^128 |
| 2 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 2 | 1 | 4 | 9.60596010000e-1 | (99/100)^4 |
| 2 | 2 | 8 | 9.22744694428e-1 | (99/100)^8 |
| 2 | 3 | 12 | 8.86384871716e-1 | (99/100)^12 |
| 2 | 4 | 16 | 8.51457771095e-1 | (99/100)^16 |
| 2 | 5 | 20 | 8.17906937597e-1 | (99/100)^20 |
| 2 | 6 | 24 | 7.85678140807e-1 | (99/100)^24 |
| 2 | 7 | 28 | 7.54719287204e-1 | (99/100)^28 |
| 2 | 8 | 32 | 7.24980335958e-1 | (99/100)^32 |
| 2 | 9 | 36 | 6.96413218050e-1 | (99/100)^36 |
| 2 | 10 | 40 | 6.68971758570e-1 | (99/100)^40 |
| 2 | 11 | 44 | 6.42611602085e-1 | (99/100)^44 |
| 2 | 12 | 48 | 6.17290140942e-1 | (99/100)^48 |
| 2 | 13 | 52 | 5.92966446401e-1 | (99/100)^52 |
| 2 | 14 | 56 | 5.69601202477e-1 | (99/100)^56 |
| 2 | 15 | 60 | 5.47156642391e-1 | (99/100)^60 |
| 2 | 16 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 2 | 17 | 68 | 5.04885888787e-1 | (99/100)^68 |
| 2 | 18 | 72 | 4.84991370274e-1 | (99/100)^72 |
| 2 | 19 | 76 | 4.65880775170e-1 | (99/100)^76 |
| 2 | 20 | 80 | 4.47523213764e-1 | (99/100)^80 |
| 2 | 21 | 84 | 4.29889013524e-1 | (99/100)^84 |
| 2 | 22 | 88 | 4.12949671134e-1 | (99/100)^88 |
| 2 | 23 | 92 | 3.96677806422e-1 | (99/100)^92 |
| 2 | 24 | 96 | 3.81047118105e-1 | (99/100)^96 |
| 2 | 25 | 100 | 3.66032341273e-1 | (99/100)^100 |
| 2 | 26 | 104 | 3.51609206558e-1 | (99/100)^104 |
| 2 | 27 | 108 | 3.37754400899e-1 | (99/100)^108 |
| 2 | 28 | 112 | 3.24445529863e-1 | (99/100)^112 |
| 2 | 29 | 116 | 3.11661081449e-1 | (99/100)^116 |
| 2 | 30 | 120 | 2.99380391312e-1 | (99/100)^120 |
| 2 | 31 | 124 | 2.87583609367e-1 | (99/100)^124 |
| 2 | 32 | 128 | 2.76251667699e-1 | (99/100)^128 |
| 2 | 33 | 132 | 2.65366249748e-1 | (99/100)^132 |
| 2 | 34 | 136 | 2.54909760696e-1 | (99/100)^136 |
| 2 | 35 | 140 | 2.44865299035e-1 | (99/100)^140 |
| 2 | 36 | 144 | 2.35216629240e-1 | (99/100)^144 |
| 2 | 37 | 148 | 2.25948155534e-1 | (99/100)^148 |
| 2 | 38 | 152 | 2.17044896673e-1 | (99/100)^152 |
| 2 | 39 | 156 | 2.08492461735e-1 | (99/100)^156 |
| 2 | 40 | 160 | 2.00277026857e-1 | (99/100)^160 |
| 2 | 41 | 164 | 1.92385312894e-1 | (99/100)^164 |
| 2 | 42 | 168 | 1.84804563949e-1 | (99/100)^168 |
| 2 | 43 | 172 | 1.77522526759e-1 | (99/100)^172 |
| 2 | 44 | 176 | 1.70527430890e-1 | (99/100)^176 |
| 2 | 45 | 180 | 1.63807969708e-1 | (99/100)^180 |
| 2 | 46 | 184 | 1.57353282108e-1 | (99/100)^184 |
| 2 | 47 | 188 | 1.51152934953e-1 | (99/100)^188 |
| 2 | 48 | 192 | 1.45196906216e-1 | (99/100)^192 |
| 2 | 49 | 196 | 1.39475568775e-1 | (99/100)^196 |
| 2 | 50 | 200 | 1.33979674858e-1 | (99/100)^200 |
| 2 | 51 | 204 | 1.28700341090e-1 | (99/100)^204 |
| 2 | 52 | 208 | 1.23629034136e-1 | (99/100)^208 |
| 2 | 53 | 212 | 1.18757556912e-1 | (99/100)^212 |
| 2 | 54 | 216 | 1.14078035327e-1 | (99/100)^216 |
| 2 | 55 | 220 | 1.09582905563e-1 | (99/100)^220 |
| 2 | 56 | 224 | 1.05264901848e-1 | (99/100)^224 |
| 2 | 57 | 228 | 1.01117044709e-1 | (99/100)^228 |
| 2 | 58 | 232 | 9.71326296900e-2 | (99/100)^232 |
| 2 | 59 | 236 | 9.33052165211e-2 | (99/100)^236 |
| 2 | 60 | 240 | 8.96286187023e-2 | (99/100)^240 |
| 2 | 61 | 244 | 8.60968935073e-2 | (99/100)^244 |
| 2 | 62 | 248 | 8.27043323765e-2 | (99/100)^248 |
| 2 | 63 | 252 | 7.94454516906e-2 | (99/100)^252 |
| 2 | 64 | 256 | 7.63149839066e-2 | (99/100)^256 |
| 3 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 3 | 1 | 8 | 9.22744694428e-1 | (99/100)^8 |
| 3 | 2 | 16 | 8.51457771095e-1 | (99/100)^16 |
| 3 | 3 | 24 | 7.85678140807e-1 | (99/100)^24 |
| 3 | 4 | 32 | 7.24980335958e-1 | (99/100)^32 |
| 3 | 5 | 40 | 6.68971758570e-1 | (99/100)^40 |
| 3 | 6 | 48 | 6.17290140942e-1 | (99/100)^48 |
| 3 | 7 | 56 | 5.69601202477e-1 | (99/100)^56 |
| 3 | 8 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 3 | 9 | 72 | 4.84991370274e-1 | (99/100)^72 |
| 3 | 10 | 80 | 4.47523213764e-1 | (99/100)^80 |
| 3 | 11 | 88 | 4.12949671134e-1 | (99/100)^88 |
| 3 | 12 | 96 | 3.81047118105e-1 | (99/100)^96 |
| 3 | 13 | 104 | 3.51609206558e-1 | (99/100)^104 |
| 3 | 14 | 112 | 3.24445529863e-1 | (99/100)^112 |
| 3 | 15 | 120 | 2.99380391312e-1 | (99/100)^120 |
| 3 | 16 | 128 | 2.76251667699e-1 | (99/100)^128 |
| 3 | 17 | 136 | 2.54909760696e-1 | (99/100)^136 |
| 3 | 18 | 144 | 2.35216629240e-1 | (99/100)^144 |
| 3 | 19 | 152 | 2.17044896673e-1 | (99/100)^152 |
| 3 | 20 | 160 | 2.00277026857e-1 | (99/100)^160 |
| 3 | 21 | 168 | 1.84804563949e-1 | (99/100)^168 |
| 3 | 22 | 176 | 1.70527430890e-1 | (99/100)^176 |
| 3 | 23 | 184 | 1.57353282108e-1 | (99/100)^184 |
| 3 | 24 | 192 | 1.45196906216e-1 | (99/100)^192 |
| 3 | 25 | 200 | 1.33979674858e-1 | (99/100)^200 |
| 3 | 26 | 208 | 1.23629034136e-1 | (99/100)^208 |
| 3 | 27 | 216 | 1.14078035327e-1 | (99/100)^216 |
| 3 | 28 | 224 | 1.05264901848e-1 | (99/100)^224 |
| 3 | 29 | 232 | 9.71326296900e-2 | (99/100)^232 |
| 3 | 30 | 240 | 8.96286187023e-2 | (99/100)^240 |
| 3 | 31 | 248 | 8.27043323765e-2 | (99/100)^248 |
| 3 | 32 | 256 | 7.63149839066e-2 | (99/100)^256 |
| 3 | 33 | 264 | 7.04192465052e-2 | (99/100)^264 |
| 3 | 34 | 272 | 6.49789860982e-2 | (99/100)^272 |
| 3 | 35 | 280 | 5.99590146715e-2 | (99/100)^280 |
| 3 | 36 | 288 | 5.53268626712e-2 | (99/100)^288 |
| 3 | 37 | 296 | 5.10525689892e-2 | (99/100)^296 |
| 3 | 38 | 304 | 4.71084871717e-2 | (99/100)^304 |
| 3 | 39 | 312 | 4.34691066002e-2 | (99/100)^312 |
| 3 | 40 | 320 | 4.01108874869e-2 | (99/100)^320 |
| 3 | 41 | 328 | 3.70121086173e-2 | (99/100)^328 |
| 3 | 42 | 336 | 3.41527268562e-2 | (99/100)^336 |
| 3 | 43 | 344 | 3.15142475068e-2 | (99/100)^344 |
| 3 | 44 | 352 | 2.90796046858e-2 | (99/100)^352 |
| 3 | 45 | 360 | 2.68330509399e-2 | (99/100)^360 |
| 3 | 46 | 368 | 2.47600553901e-2 | (99/100)^368 |
| 3 | 47 | 376 | 2.28472097450e-2 | (99/100)^376 |
| 3 | 48 | 384 | 2.10821415746e-2 | (99/100)^384 |
| 3 | 49 | 392 | 1.94534342852e-2 | (99/100)^392 |
| 3 | 50 | 400 | 1.79505532750e-2 | (99/100)^400 |
| 3 | 51 | 408 | 1.65637777966e-2 | (99/100)^408 |
| 3 | 52 | 416 | 1.52841380815e-2 | (99/100)^416 |
| 3 | 53 | 424 | 1.41033573236e-2 | (99/100)^424 |
| 3 | 54 | 432 | 1.30137981440e-2 | (99/100)^432 |
| 3 | 55 | 440 | 1.20084131917e-2 | (99/100)^440 |
| 3 | 56 | 448 | 1.10806995611e-2 | (99/100)^448 |
| 3 | 57 | 456 | 1.02246567306e-2 | (99/100)^456 |
| 3 | 58 | 464 | 9.43474775050e-3 | (99/100)^464 |
| 3 | 59 | 472 | 8.70586343004e-3 | (99/100)^472 |
| 3 | 60 | 480 | 8.03328929049e-3 | (99/100)^480 |
| 3 | 61 | 488 | 7.41267507160e-3 | (99/100)^488 |
| 3 | 62 | 496 | 6.84000659384e-3 | (99/100)^496 |
| 3 | 63 | 504 | 6.31157979432e-3 | (99/100)^504 |
| 3 | 64 | 512 | 5.82397676866e-3 | (99/100)^512 |
| 4 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 4 | 1 | 16 | 8.51457771095e-1 | (99/100)^16 |
| 4 | 2 | 32 | 7.24980335958e-1 | (99/100)^32 |
| 4 | 3 | 48 | 6.17290140942e-1 | (99/100)^48 |
| 4 | 4 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 4 | 5 | 80 | 4.47523213764e-1 | (99/100)^80 |
| 4 | 6 | 96 | 3.81047118105e-1 | (99/100)^96 |
| 4 | 7 | 112 | 3.24445529863e-1 | (99/100)^112 |
| 4 | 8 | 128 | 2.76251667699e-1 | (99/100)^128 |
| 4 | 9 | 144 | 2.35216629240e-1 | (99/100)^144 |
| 4 | 10 | 160 | 2.00277026857e-1 | (99/100)^160 |
| 4 | 11 | 176 | 1.70527430890e-1 | (99/100)^176 |
| 4 | 12 | 192 | 1.45196906216e-1 | (99/100)^192 |
| 4 | 13 | 208 | 1.23629034136e-1 | (99/100)^208 |
| 4 | 14 | 224 | 1.05264901848e-1 | (99/100)^224 |
| 4 | 15 | 240 | 8.96286187023e-2 | (99/100)^240 |
| 4 | 16 | 256 | 7.63149839066e-2 | (99/100)^256 |
| 4 | 17 | 272 | 6.49789860982e-2 | (99/100)^272 |
| 4 | 18 | 288 | 5.53268626712e-2 | (99/100)^288 |
| 4 | 19 | 304 | 4.71084871717e-2 | (99/100)^304 |
| 4 | 20 | 320 | 4.01108874869e-2 | (99/100)^320 |
| 4 | 21 | 336 | 3.41527268562e-2 | (99/100)^336 |
| 4 | 22 | 352 | 2.90796046858e-2 | (99/100)^352 |
| 4 | 23 | 368 | 2.47600553901e-2 | (99/100)^368 |
| 4 | 24 | 384 | 2.10821415746e-2 | (99/100)^384 |
| 4 | 25 | 400 | 1.79505532750e-2 | (99/100)^400 |
| 4 | 26 | 416 | 1.52841380815e-2 | (99/100)^416 |
| 4 | 27 | 432 | 1.30137981440e-2 | (99/100)^432 |
| 4 | 28 | 448 | 1.10806995611e-2 | (99/100)^448 |
| 4 | 29 | 464 | 9.43474775050e-3 | (99/100)^464 |
| 4 | 30 | 480 | 8.03328929049e-3 | (99/100)^480 |
| 4 | 31 | 496 | 6.84000659384e-3 | (99/100)^496 |
| 4 | 32 | 512 | 5.82397676866e-3 | (99/100)^512 |
| 4 | 33 | 528 | 4.95887027835e-3 | (99/100)^528 |
| 4 | 34 | 544 | 4.22226863436e-3 | (99/100)^544 |
| 4 | 35 | 560 | 3.59508344037e-3 | (99/100)^560 |
| 4 | 36 | 576 | 3.06106173304e-3 | (99/100)^576 |
| 4 | 37 | 592 | 2.60636480040e-3 | (99/100)^592 |
| 4 | 38 | 608 | 2.21920956361e-3 | (99/100)^608 |
| 4 | 39 | 624 | 1.88956322862e-3 | (99/100)^624 |
| 4 | 40 | 640 | 1.60888329498e-3 | (99/100)^640 |
| 4 | 41 | 656 | 1.36989618430e-3 | (99/100)^656 |
| 4 | 42 | 672 | 1.16640875172e-3 | (99/100)^672 |
| 4 | 43 | 688 | 9.93147795921e-4 | (99/100)^688 |
| 4 | 44 | 704 | 8.45623408683e-4 | (99/100)^704 |
| 4 | 45 | 720 | 7.20012622743e-4 | (99/100)^720 |
| 4 | 46 | 736 | 6.13060342921e-4 | (99/100)^736 |
| 4 | 47 | 752 | 5.21994993130e-4 | (99/100)^752 |
| 4 | 48 | 768 | 4.44456693373e-4 | (99/100)^768 |
| 4 | 49 | 784 | 3.78436105488e-4 | (99/100)^784 |
| 4 | 50 | 800 | 3.22222362880e-4 | (99/100)^800 |
| 4 | 51 | 816 | 2.74358734895e-4 | (99/100)^816 |
| 4 | 52 | 832 | 2.33604876894e-4 | (99/100)^832 |
| 4 | 53 | 848 | 1.98904687797e-4 | (99/100)^848 |
| 4 | 54 | 864 | 1.69358942132e-4 | (99/100)^864 |
| 4 | 55 | 880 | 1.44201987383e-4 | (99/100)^880 |
| 4 | 56 | 896 | 1.22781902764e-4 | (99/100)^896 |
| 4 | 57 | 912 | 1.04543605259e-4 | (99/100)^912 |
| 4 | 58 | 928 | 8.90144651156e-5 | (99/100)^928 |
| 4 | 59 | 944 | 7.57920580626e-5 | (99/100)^944 |
| 4 | 60 | 960 | 6.45337368246e-5 | (99/100)^960 |
| 4 | 61 | 976 | 5.49477517171e-5 | (99/100)^976 |
| 4 | 62 | 992 | 4.67856902037e-5 | (99/100)^992 |
| 4 | 63 | 1008 | 3.98360395000e-5 | (99/100)^1008 |
| 4 | 64 | 1024 | 3.39187054019e-5 | (99/100)^1024 |
| 5 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 5 | 1 | 32 | 7.24980335958e-1 | (99/100)^32 |
| 5 | 2 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 5 | 3 | 96 | 3.81047118105e-1 | (99/100)^96 |
| 5 | 4 | 128 | 2.76251667699e-1 | (99/100)^128 |
| 5 | 5 | 160 | 2.00277026857e-1 | (99/100)^160 |
| 5 | 6 | 192 | 1.45196906216e-1 | (99/100)^192 |
| 5 | 7 | 224 | 1.05264901848e-1 | (99/100)^224 |
| 5 | 8 | 256 | 7.63149839066e-2 | (99/100)^256 |
| 5 | 9 | 288 | 5.53268626712e-2 | (99/100)^288 |
| 5 | 10 | 320 | 4.01108874869e-2 | (99/100)^320 |
| 5 | 11 | 352 | 2.90796046858e-2 | (99/100)^352 |
| 5 | 12 | 384 | 2.10821415746e-2 | (99/100)^384 |
| 5 | 13 | 416 | 1.52841380815e-2 | (99/100)^416 |
| 5 | 14 | 448 | 1.10806995611e-2 | (99/100)^448 |
| 5 | 15 | 480 | 8.03328929049e-3 | (99/100)^480 |
| 5 | 16 | 512 | 5.82397676866e-3 | (99/100)^512 |
| 5 | 17 | 544 | 4.22226863436e-3 | (99/100)^544 |
| 5 | 18 | 576 | 3.06106173304e-3 | (99/100)^576 |
| 5 | 19 | 608 | 2.21920956361e-3 | (99/100)^608 |
| 5 | 20 | 640 | 1.60888329498e-3 | (99/100)^640 |
| 5 | 21 | 672 | 1.16640875172e-3 | (99/100)^672 |
| 5 | 22 | 704 | 8.45623408683e-4 | (99/100)^704 |
| 5 | 23 | 736 | 6.13060342921e-4 | (99/100)^736 |
| 5 | 24 | 768 | 4.44456693373e-4 | (99/100)^768 |
| 5 | 25 | 800 | 3.22222362880e-4 | (99/100)^800 |
| 5 | 26 | 832 | 2.33604876894e-4 | (99/100)^832 |
| 5 | 27 | 864 | 1.69358942132e-4 | (99/100)^864 |
| 5 | 28 | 896 | 1.22781902764e-4 | (99/100)^896 |
| 5 | 29 | 928 | 8.90144651156e-5 | (99/100)^928 |
| 5 | 30 | 960 | 6.45337368246e-5 | (99/100)^960 |
| 5 | 31 | 992 | 4.67856902037e-5 | (99/100)^992 |
| 5 | 32 | 1024 | 3.39187054019e-5 | (99/100)^1024 |
| 5 | 33 | 1056 | 2.45903944375e-5 | (99/100)^1056 |
| 5 | 34 | 1088 | 1.78275524207e-5 | (99/100)^1088 |
| 5 | 35 | 1120 | 1.29246249432e-5 | (99/100)^1120 |
| 5 | 36 | 1152 | 9.37009893348e-6 | (99/100)^1152 |
| 5 | 37 | 1184 | 6.79313747275e-6 | (99/100)^1184 |
| 5 | 38 | 1216 | 4.92489108721e-6 | (99/100)^1216 |
| 5 | 39 | 1248 | 3.57044919496e-6 | (99/100)^1248 |
| 5 | 40 | 1280 | 2.58850545688e-6 | (99/100)^1280 |
| 5 | 41 | 1312 | 1.87661555576e-6 | (99/100)^1312 |
| 5 | 42 | 1344 | 1.36050937608e-6 | (99/100)^1344 |
| 5 | 43 | 1376 | 9.86342544542e-7 | (99/100)^1376 |
| 5 | 44 | 1408 | 7.15078949312e-7 | (99/100)^1408 |
| 5 | 45 | 1440 | 5.18418176909e-7 | (99/100)^1440 |
| 5 | 46 | 1472 | 3.75842984062e-7 | (99/100)^1472 |
| 5 | 47 | 1504 | 2.72478772853e-7 | (99/100)^1504 |
| 5 | 48 | 1536 | 1.97541752284e-7 | (99/100)^1536 |
| 5 | 49 | 1568 | 1.43213885937e-7 | (99/100)^1568 |
| 5 | 50 | 1600 | 1.03827251140e-7 | (99/100)^1600 |
| 5 | 51 | 1632 | 7.52727154131e-8 | (99/100)^1632 |
| 5 | 52 | 1664 | 5.45712385087e-8 | (99/100)^1664 |
| 5 | 53 | 1696 | 3.95630748277e-8 | (99/100)^1696 |
| 5 | 54 | 1728 | 2.86824512801e-8 | (99/100)^1728 |
| 5 | 55 | 1760 | 2.07942131651e-8 | (99/100)^1760 |
| 5 | 56 | 1792 | 1.50753956464e-8 | (99/100)^1792 |
| 5 | 57 | 1824 | 1.09293654004e-8 | (99/100)^1824 |
| 5 | 58 | 1856 | 7.92357499982e-9 | (99/100)^1856 |
| 5 | 59 | 1888 | 5.74443606536e-9 | (99/100)^1888 |
| 5 | 60 | 1920 | 4.16460318855e-9 | (99/100)^1920 |
| 5 | 61 | 1952 | 3.01925541877e-9 | (99/100)^1952 |
| 5 | 62 | 1984 | 2.18890080784e-9 | (99/100)^1984 |
| 5 | 63 | 2016 | 1.58691004305e-9 | (99/100)^2016 |
| 5 | 64 | 2048 | 1.15047857614e-9 | (99/100)^2048 |
| 6 | 0 | 0 | 1.00000000000e+0 | (99/100)^0 |
| 6 | 1 | 64 | 5.25596487526e-1 | (99/100)^64 |
| 6 | 2 | 128 | 2.76251667699e-1 | (99/100)^128 |
| 6 | 3 | 192 | 1.45196906216e-1 | (99/100)^192 |
| 6 | 4 | 256 | 7.63149839066e-2 | (99/100)^256 |
| 6 | 5 | 320 | 4.01108874869e-2 | (99/100)^320 |
| 6 | 6 | 384 | 2.10821415746e-2 | (99/100)^384 |
| 6 | 7 | 448 | 1.10806995611e-2 | (99/100)^448 |
| 6 | 8 | 512 | 5.82397676866e-3 | (99/100)^512 |
| 6 | 9 | 576 | 3.06106173304e-3 | (99/100)^576 |
| 6 | 10 | 640 | 1.60888329498e-3 | (99/100)^640 |
| 6 | 11 | 704 | 8.45623408683e-4 | (99/100)^704 |
| 6 | 12 | 768 | 4.44456693373e-4 | (99/100)^768 |
| 6 | 13 | 832 | 2.33604876894e-4 | (99/100)^832 |
| 6 | 14 | 896 | 1.22781902764e-4 | (99/100)^896 |
| 6 | 15 | 960 | 6.45337368246e-5 | (99/100)^960 |
| 6 | 16 | 1024 | 3.39187054019e-5 | (99/100)^1024 |
| 6 | 17 | 1088 | 1.78275524207e-5 | (99/100)^1088 |
| 6 | 18 | 1152 | 9.37009893348e-6 | (99/100)^1152 |
| 6 | 19 | 1216 | 4.92489108721e-6 | (99/100)^1216 |
| 6 | 20 | 1280 | 2.58850545688e-6 | (99/100)^1280 |
| 6 | 21 | 1344 | 1.36050937608e-6 | (99/100)^1344 |
| 6 | 22 | 1408 | 7.15078949312e-7 | (99/100)^1408 |
| 6 | 23 | 1472 | 3.75842984062e-7 | (99/100)^1472 |
| 6 | 24 | 1536 | 1.97541752284e-7 | (99/100)^1536 |
| 6 | 25 | 1600 | 1.03827251140e-7 | (99/100)^1600 |
| 6 | 26 | 1664 | 5.45712385087e-8 | (99/100)^1664 |
| 6 | 27 | 1728 | 2.86824512801e-8 | (99/100)^1728 |
| 6 | 28 | 1792 | 1.50753956464e-8 | (99/100)^1792 |
| 6 | 29 | 1856 | 7.92357499982e-9 | (99/100)^1856 |
| 6 | 30 | 1920 | 4.16460318855e-9 | (99/100)^1920 |
| 6 | 31 | 1984 | 2.18890080784e-9 | (99/100)^1984 |
| 6 | 32 | 2048 | 1.15047857614e-9 | (99/100)^2048 |
| 6 | 33 | 2112 | 6.04687498594e-10 | (99/100)^2112 |
| 6 | 34 | 2176 | 3.17821625312e-10 | (99/100)^2176 |
| 6 | 35 | 2240 | 1.67045929924e-10 | (99/100)^2240 |
| 6 | 36 | 2304 | 8.77987540233e-11 | (99/100)^2304 |
| 6 | 37 | 2368 | 4.61467167237e-11 | (99/100)^2368 |
| 6 | 38 | 2432 | 2.42545522208e-11 | (99/100)^2432 |
| 6 | 39 | 2496 | 1.27481074538e-11 | (99/100)^2496 |
| 6 | 40 | 2560 | 6.70036050030e-12 | (99/100)^2560 |
| 6 | 41 | 2624 | 3.52168594411e-12 | (99/100)^2624 |
| 6 | 42 | 2688 | 1.85098576239e-12 | (99/100)^2688 |
| 6 | 43 | 2752 | 9.72871615174e-13 | (99/100)^2752 |
| 6 | 44 | 2816 | 5.11337903749e-13 | (99/100)^2816 |
| 6 | 45 | 2880 | 2.68757406149e-13 | (99/100)^2880 |
| 6 | 46 | 2944 | 1.41257948668e-13 | (99/100)^2944 |
| 6 | 47 | 3008 | 7.42446816552e-14 | (99/100)^3008 |
| 6 | 48 | 3072 | 3.90227438954e-14 | (99/100)^3072 |
| 6 | 49 | 3136 | 2.05102171251e-14 | (99/100)^3136 |
| 6 | 50 | 3200 | 1.07800980793e-14 | (99/100)^3200 |
| 6 | 51 | 3264 | 5.66598168567e-15 | (99/100)^3264 |
| 6 | 52 | 3328 | 2.97802007237e-15 | (99/100)^3328 |
| 6 | 53 | 3392 | 1.56523688982e-15 | (99/100)^3392 |
| 6 | 54 | 3456 | 8.22683011434e-16 | (99/100)^3456 |
| 6 | 55 | 3520 | 4.32399301157e-16 | (99/100)^3520 |
| 6 | 56 | 3584 | 2.27267553897e-16 | (99/100)^3584 |
| 6 | 57 | 3648 | 1.19451028057e-16 | (99/100)^3648 |
| 6 | 58 | 3712 | 6.27830407778e-17 | (99/100)^3712 |
| 6 | 59 | 3776 | 3.29985457090e-17 | (99/100)^3776 |
| 6 | 60 | 3840 | 1.73439197181e-17 | (99/100)^3840 |
| 6 | 61 | 3904 | 9.11590328376e-18 | (99/100)^3904 |
| 6 | 62 | 3968 | 4.79128674657e-18 | (99/100)^3968 |
| 6 | 63 | 4032 | 2.51828348472e-18 | (99/100)^4032 |
| 6 | 64 | 4096 | 1.32360095416e-18 | (99/100)^4096 |
On a narrow screen, focus or swipe within a figure or table to scroll horizontally. Labels retain their designed size.
Separate coarse-graining, rescaling and field normalization
A general spatial renormalization step has three logically different parts. First sum or integrate short-distance degrees of freedom while retaining their effect on the remaining statistical weights. Next express lengths in coarser units, x′=x/b with b>1. Finally choose a normalization for the retained field so that descriptions at successive scales can be compared.
In a continuum field model, one can integrate a shell of large wave numbers, Λ/b<|k|<Λ, where Λ is the ultraviolet cutoff. The remaining effective action generally contains many interactions. Restoring the cutoff by coordinate rescaling and normalizing the coefficient of (∇φ)² defines a flow of their couplings. Real-space blocking and momentum-shell integration are different implementations of the same scale-comparison idea, not interchangeable numerical algorithms.
Coarse-graining need not preserve the appearance of a single configuration. What is compared is its probability law and appropriate long-distance observables. The transformation is not generally invertible once short-distance information has been summed out; calling it a group does not give a recipe for recovering every removed microscopic detail.
Identify fixed points and relevant directions
A fixed point is a statistical theory whose dimensionless couplings return to the same values after the full declared scale transformation. Close to it, suitable combinations of perturbations behave as g_i(b)≈b^(y_i)g_i. These combinations are scaling fields; an original laboratory temperature or coupling need not already be one of them.
With our convention of increasing b toward longer distances, y_i>0 is relevant: the perturbation grows. A negative y_i is irrelevant and decays locally. A zero y_i is marginal at this order; nonlinear terms decide whether it grows, decays or remains exactly marginal. Reversing the flow convention reverses how a beta function is written, so its sign must not be interpreted without its convention.
Near many ordinary scalar critical points, temperature-like and symmetry-breaking field-like directions are relevant. Reaching the critical point requires tuning the appropriate relevant variables. Microscopic changes along irrelevant directions can become unimportant for leading long-distance behaviour, but can still produce finite-scale corrections or affect amplitudes. Irrelevant does not mean experimentally nonexistent.
See a growing length in a static Gaussian model
Consider a separate assigned Gaussian static structure factor S(k)=1/[r+(ka)²], with positive dimensionless r, wave number k and reference length a. We have chosen its amplitude and gradient coefficient to be 1. This model describes equilibrium spatial fluctuations; S is neither a normalized probability density nor a frequency-dependent noise spectrum.
Its zero-wave-number value is S(0)=1/r, and its correlation length is ξ/a=1/√r. At kξ=1 the structure factor is half its zero-wave-number value. Decreasing r therefore produces a taller, narrower peak in k while increasing the associated length in real space. The reciprocal relation concerns a width and a length, not an assertion that every fluctuation has a single wavelength.
Divide the vertical axis by S(0) and express the horizontal axis as kξ. Every supplied curve obeys S(k)/S(0)=1/[1+(kξ)²]. This collapse follows exactly from the assigned equation. It is not a fitted measurement of a universality class. The supplied equal ka grids map to different kξ ranges; comparison must use their actual coordinates rather than stretching all endpoints to coincide.
A Gaussian description can be appropriate away from an interacting critical region or above the appropriate upper critical dimension. Close enough to a non-Gaussian critical point, the neglected interactions change the long-distance behaviour. A growing measured correlation length is an invitation to test that regime, not automatic permission to extrapolate the Gaussian exponent 1/2 to every material.
A narrower peak, a longer length
The three assigned static Gaussian models share an equation, not measured data. First compare their raw peaks, then compare their normalized shapes on the actual scaled coordinates.
Raw structure factors
- r=1/25
- r=4/25
- r=1
One shape, three different horizontal reaches
- r=1/25; kξ ends at 25
- r=4/25; kξ ends at 12.5
- r=1; kξ ends at 5
- The scaled curves intentionally overlap; a shared shape does not mean their sampled endpoints are equal. No curve is stretched to another endpoint.
- This exact assigned collapse is not an empirical fit or proof of an interacting universality class. It is a static spatial structure factor, not a probability density or a frequency spectrum.
The original whole-table links and complete additional values are below.
Every assigned Gaussian sample for r=1/25. Rounded to 12 significant digits; kξ uses the actual ξ/a, so scaled endpoints differ between controls.
| ka | kξ | S(k)≈ | S(k)/S(0)≈ |
|---|---|---|---|
| 0 | 0 | 2.50000000000e+1 | 1.00000000000e+0 |
| 2.50000000000e-2 | 1.25000000000e-1 | 2.46153846154e+1 | 9.84615384615e-1 |
| 5.00000000000e-2 | 2.50000000000e-1 | 2.35294117647e+1 | 9.41176470588e-1 |
| 7.50000000000e-2 | 3.75000000000e-1 | 2.19178082192e+1 | 8.76712328767e-1 |
| 1.00000000000e-1 | 5.00000000000e-1 | 2.00000000000e+1 | 8.00000000000e-1 |
| 1.25000000000e-1 | 6.25000000000e-1 | 1.79775280899e+1 | 7.19101123596e-1 |
| 1.50000000000e-1 | 7.50000000000e-1 | 1.60000000000e+1 | 6.40000000000e-1 |
| 1.75000000000e-1 | 8.75000000000e-1 | 1.41592920354e+1 | 5.66371681416e-1 |
| 2.00000000000e-1 | 1.00000000000e+0 | 1.25000000000e+1 | 5.00000000000e-1 |
| 2.25000000000e-1 | 1.12500000000e+0 | 1.10344827586e+1 | 4.41379310345e-1 |
| 2.50000000000e-1 | 1.25000000000e+0 | 9.75609756098e+0 | 3.90243902439e-1 |
| 2.75000000000e-1 | 1.37500000000e+0 | 8.64864864865e+0 | 3.45945945946e-1 |
| 3.00000000000e-1 | 1.50000000000e+0 | 7.69230769231e+0 | 3.07692307692e-1 |
| 3.25000000000e-1 | 1.62500000000e+0 | 6.86695278970e+0 | 2.74678111588e-1 |
| 3.50000000000e-1 | 1.75000000000e+0 | 6.15384615385e+0 | 2.46153846154e-1 |
| 3.75000000000e-1 | 1.87500000000e+0 | 5.53633217993e+0 | 2.21453287197e-1 |
| 4.00000000000e-1 | 2.00000000000e+0 | 5.00000000000e+0 | 2.00000000000e-1 |
| 4.25000000000e-1 | 2.12500000000e+0 | 4.53257790368e+0 | 1.81303116147e-1 |
| 4.50000000000e-1 | 2.25000000000e+0 | 4.12371134021e+0 | 1.64948453608e-1 |
| 4.75000000000e-1 | 2.37500000000e+0 | 3.76470588235e+0 | 1.50588235294e-1 |
| 5.00000000000e-1 | 2.50000000000e+0 | 3.44827586207e+0 | 1.37931034483e-1 |
| 5.25000000000e-1 | 2.62500000000e+0 | 3.16831683168e+0 | 1.26732673267e-1 |
| 5.50000000000e-1 | 2.75000000000e+0 | 2.91970802920e+0 | 1.16788321168e-1 |
| 5.75000000000e-1 | 2.87500000000e+0 | 2.69814502530e+0 | 1.07925801012e-1 |
| 6.00000000000e-1 | 3.00000000000e+0 | 2.50000000000e+0 | 1.00000000000e-1 |
| 6.25000000000e-1 | 3.12500000000e+0 | 2.32220609579e+0 | 9.28882438316e-2 |
| 6.50000000000e-1 | 3.25000000000e+0 | 2.16216216216e+0 | 8.64864864865e-2 |
| 6.75000000000e-1 | 3.37500000000e+0 | 2.01765447667e+0 | 8.07061790668e-2 |
| 7.00000000000e-1 | 3.50000000000e+0 | 1.88679245283e+0 | 7.54716981132e-2 |
| 7.25000000000e-1 | 3.62500000000e+0 | 1.76795580110e+0 | 7.07182320442e-2 |
| 7.50000000000e-1 | 3.75000000000e+0 | 1.65975103734e+0 | 6.63900414938e-2 |
| 7.75000000000e-1 | 3.87500000000e+0 | 1.56097560976e+0 | 6.24390243902e-2 |
| 8.00000000000e-1 | 4.00000000000e+0 | 1.47058823529e+0 | 5.88235294118e-2 |
| 8.25000000000e-1 | 4.12500000000e+0 | 1.38768430182e+0 | 5.55073720729e-2 |
| 8.50000000000e-1 | 4.25000000000e+0 | 1.31147540984e+0 | 5.24590163934e-2 |
| 8.75000000000e-1 | 4.37500000000e+0 | 1.24127230411e+0 | 4.96508921645e-2 |
| 9.00000000000e-1 | 4.50000000000e+0 | 1.17647058824e+0 | 4.70588235294e-2 |
| 9.25000000000e-1 | 4.62500000000e+0 | 1.11653872994e+0 | 4.46615491975e-2 |
| 9.50000000000e-1 | 4.75000000000e+0 | 1.06100795756e+0 | 4.24403183024e-2 |
| 9.75000000000e-1 | 4.87500000000e+0 | 1.00946372240e+0 | 4.03785488959e-2 |
| 1.00000000000e+0 | 5.00000000000e+0 | 9.61538461538e-1 | 3.84615384615e-2 |
| 1.02500000000e+0 | 5.12500000000e+0 | 9.16905444126e-1 | 3.66762177650e-2 |
| 1.05000000000e+0 | 5.25000000000e+0 | 8.75273522976e-1 | 3.50109409190e-2 |
| 1.07500000000e+0 | 5.37500000000e+0 | 8.36382645060e-1 | 3.34553058024e-2 |
| 1.10000000000e+0 | 5.50000000000e+0 | 8.00000000000e-1 | 3.20000000000e-2 |
| 1.12500000000e+0 | 5.62500000000e+0 | 7.65916706558e-1 | 3.06366682623e-2 |
| 1.15000000000e+0 | 5.75000000000e+0 | 7.33944954128e-1 | 2.93577981651e-2 |
| 1.17500000000e+0 | 5.87500000000e+0 | 7.03915530136e-1 | 2.81566212055e-2 |
| 1.20000000000e+0 | 6.00000000000e+0 | 6.75675675676e-1 | 2.70270270270e-2 |
| 1.22500000000e+0 | 6.12500000000e+0 | 6.49087221095e-1 | 2.59634888438e-2 |
| 1.25000000000e+0 | 6.25000000000e+0 | 6.24024960998e-1 | 2.49609984399e-2 |
| 1.27500000000e+0 | 6.37500000000e+0 | 6.00375234522e-1 | 2.40150093809e-2 |
| 1.30000000000e+0 | 6.50000000000e+0 | 5.78034682081e-1 | 2.31213872832e-2 |
| 1.32500000000e+0 | 6.62500000000e+0 | 5.56909154194e-1 | 2.22763661678e-2 |
| 1.35000000000e+0 | 6.75000000000e+0 | 5.36912751678e-1 | 2.14765100671e-2 |
| 1.37500000000e+0 | 6.87500000000e+0 | 5.17966979605e-1 | 2.07186791842e-2 |
| 1.40000000000e+0 | 7.00000000000e+0 | 5.00000000000e-1 | 2.00000000000e-2 |
| 1.42500000000e+0 | 7.12500000000e+0 | 4.82945970420e-1 | 1.93178388168e-2 |
| 1.45000000000e+0 | 7.25000000000e+0 | 4.66744457410e-1 | 1.86697782964e-2 |
| 1.47500000000e+0 | 7.37500000000e+0 | 4.51339915374e-1 | 1.80535966150e-2 |
| 1.50000000000e+0 | 7.50000000000e+0 | 4.36681222707e-1 | 1.74672489083e-2 |
| 1.52500000000e+0 | 7.62500000000e+0 | 4.22721268164e-1 | 1.69088507266e-2 |
| 1.55000000000e+0 | 7.75000000000e+0 | 4.09416581372e-1 | 1.63766632549e-2 |
| 1.57500000000e+0 | 7.87500000000e+0 | 3.96727002232e-1 | 1.58690800893e-2 |
| 1.60000000000e+0 | 8.00000000000e+0 | 3.84615384615e-1 | 1.53846153846e-2 |
| 1.62500000000e+0 | 8.12500000000e+0 | 3.73047330380e-1 | 1.49218932152e-2 |
| 1.65000000000e+0 | 8.25000000000e+0 | 3.61990950226e-1 | 1.44796380090e-2 |
| 1.67500000000e+0 | 8.37500000000e+0 | 3.51416648364e-1 | 1.40566659345e-2 |
| 1.70000000000e+0 | 8.50000000000e+0 | 3.41296928328e-1 | 1.36518771331e-2 |
| 1.72500000000e+0 | 8.62500000000e+0 | 3.31606217617e-1 | 1.32642487047e-2 |
| 1.75000000000e+0 | 8.75000000000e+0 | 3.22320709106e-1 | 1.28928283642e-2 |
| 1.77500000000e+0 | 8.87500000000e+0 | 3.13418217434e-1 | 1.25367286974e-2 |
| 1.80000000000e+0 | 9.00000000000e+0 | 3.04878048780e-1 | 1.21951219512e-2 |
| 1.82500000000e+0 | 9.12500000000e+0 | 2.96680882626e-1 | 1.18672353050e-2 |
| 1.85000000000e+0 | 9.25000000000e+0 | 2.88808664260e-1 | 1.15523465704e-2 |
| 1.87500000000e+0 | 9.37500000000e+0 | 2.81244506943e-1 | 1.12497802777e-2 |
| 1.90000000000e+0 | 9.50000000000e+0 | 2.73972602740e-1 | 1.09589041096e-2 |
| 1.92500000000e+0 | 9.62500000000e+0 | 2.66978141165e-1 | 1.06791256466e-2 |
| 1.95000000000e+0 | 9.75000000000e+0 | 2.60247234873e-1 | 1.04098893949e-2 |
| 1.97500000000e+0 | 9.87500000000e+0 | 2.53766851705e-1 | 1.01506740682e-2 |
| 2.00000000000e+0 | 1.00000000000e+1 | 2.47524752475e-1 | 9.90099009901e-3 |
| 2.02500000000e+0 | 1.01250000000e+1 | 2.41509433962e-1 | 9.66037735849e-3 |
| 2.05000000000e+0 | 1.02500000000e+1 | 2.35710076606e-1 | 9.42840306423e-3 |
| 2.07500000000e+0 | 1.03750000000e+1 | 2.30116496476e-1 | 9.20465985905e-3 |
| 2.10000000000e+0 | 1.05000000000e+1 | 2.24719101124e-1 | 8.98876404494e-3 |
| 2.12500000000e+0 | 1.06250000000e+1 | 2.19508848950e-1 | 8.78035395802e-3 |
| 2.15000000000e+0 | 1.07500000000e+1 | 2.14477211796e-1 | 8.57908847185e-3 |
| 2.17500000000e+0 | 1.08750000000e+1 | 2.09616140443e-1 | 8.38464561771e-3 |
| 2.20000000000e+0 | 1.10000000000e+1 | 2.04918032787e-1 | 8.19672131148e-3 |
| 2.22500000000e+0 | 1.11250000000e+1 | 2.00375704446e-1 | 8.01502817783e-3 |
| 2.25000000000e+0 | 1.12500000000e+1 | 1.95982361587e-1 | 7.83929446350e-3 |
| 2.27500000000e+0 | 1.13750000000e+1 | 1.91731575794e-1 | 7.66926303176e-3 |
| 2.30000000000e+0 | 1.15000000000e+1 | 1.87617260788e-1 | 7.50469043152e-3 |
| 2.32500000000e+0 | 1.16250000000e+1 | 1.83633650867e-1 | 7.34534603466e-3 |
| 2.35000000000e+0 | 1.17500000000e+1 | 1.79775280899e-1 | 7.19101123596e-3 |
| 2.37500000000e+0 | 1.18750000000e+1 | 1.76036967763e-1 | 7.04147871053e-3 |
| 2.40000000000e+0 | 1.20000000000e+1 | 1.72413793103e-1 | 6.89655172414e-3 |
| 2.42500000000e+0 | 1.21250000000e+1 | 1.68901087301e-1 | 6.75604349203e-3 |
| 2.45000000000e+0 | 1.22500000000e+1 | 1.65494414564e-1 | 6.61977658254e-3 |
| 2.47500000000e+0 | 1.23750000000e+1 | 1.62189559047e-1 | 6.48758236189e-3 |
| 2.50000000000e+0 | 1.25000000000e+1 | 1.58982511924e-1 | 6.35930047695e-3 |
| 2.52500000000e+0 | 1.26250000000e+1 | 1.55869459328e-1 | 6.23477837311e-3 |
| 2.55000000000e+0 | 1.27500000000e+1 | 1.52846771112e-1 | 6.11387084448e-3 |
| 2.57500000000e+0 | 1.28750000000e+1 | 1.49910990349e-1 | 5.99643961398e-3 |
| 2.60000000000e+0 | 1.30000000000e+1 | 1.47058823529e-1 | 5.88235294118e-3 |
| 2.62500000000e+0 | 1.31250000000e+1 | 1.44287131391e-1 | 5.77148525566e-3 |
| 2.65000000000e+0 | 1.32500000000e+1 | 1.41592920354e-1 | 5.66371681416e-3 |
| 2.67500000000e+0 | 1.33750000000e+1 | 1.38973334491e-1 | 5.55893337966e-3 |
| 2.70000000000e+0 | 1.35000000000e+1 | 1.36425648022e-1 | 5.45702592087e-3 |
| 2.72500000000e+0 | 1.36250000000e+1 | 1.33947258267e-1 | 5.35789033068e-3 |
| 2.75000000000e+0 | 1.37500000000e+1 | 1.31535679053e-1 | 5.26142716212e-3 |
| 2.77500000000e+0 | 1.38750000000e+1 | 1.29188534518e-1 | 5.16754138070e-3 |
| 2.80000000000e+0 | 1.40000000000e+1 | 1.26903553299e-1 | 5.07614213198e-3 |
| 2.82500000000e+0 | 1.41250000000e+1 | 1.24678563080e-1 | 4.98714252318e-3 |
| 2.85000000000e+0 | 1.42500000000e+1 | 1.22511485452e-1 | 4.90045941807e-3 |
| 2.87500000000e+0 | 1.43750000000e+1 | 1.20400331101e-1 | 4.81601324404e-3 |
| 2.90000000000e+0 | 1.45000000000e+1 | 1.18343195266e-1 | 4.73372781065e-3 |
| 2.92500000000e+0 | 1.46250000000e+1 | 1.16338253472e-1 | 4.65353013888e-3 |
| 2.95000000000e+0 | 1.47500000000e+1 | 1.14383757506e-1 | 4.57535030026e-3 |
| 2.97500000000e+0 | 1.48750000000e+1 | 1.12478031634e-1 | 4.49912126538e-3 |
| 3.00000000000e+0 | 1.50000000000e+1 | 1.10619469027e-1 | 4.42477876106e-3 |
| 3.02500000000e+0 | 1.51250000000e+1 | 1.08806528392e-1 | 4.35226113567e-3 |
| 3.05000000000e+0 | 1.52500000000e+1 | 1.07037730800e-1 | 4.28150923200e-3 |
| 3.07500000000e+0 | 1.53750000000e+1 | 1.05311656684e-1 | 4.21246626736e-3 |
| 3.10000000000e+0 | 1.55000000000e+1 | 1.03626943005e-1 | 4.14507772021e-3 |
| 3.12500000000e+0 | 1.56250000000e+1 | 1.01982280579e-1 | 4.07929122315e-3 |
| 3.15000000000e+0 | 1.57500000000e+1 | 1.00376411543e-1 | 4.01505646173e-3 |
| 3.17500000000e+0 | 1.58750000000e+1 | 9.88081269684e-2 | 3.95232507874e-3 |
| 3.20000000000e+0 | 1.60000000000e+1 | 9.72762645914e-2 | 3.89105058366e-3 |
| 3.22500000000e+0 | 1.61250000000e+1 | 9.57797066746e-2 | 3.83118826699e-3 |
| 3.25000000000e+0 | 1.62500000000e+1 | 9.43173779769e-2 | 3.77269511908e-3 |
| 3.27500000000e+0 | 1.63750000000e+1 | 9.28882438316e-2 | 3.71552975327e-3 |
| 3.30000000000e+0 | 1.65000000000e+1 | 9.14913083257e-2 | 3.65965233303e-3 |
| 3.32500000000e+0 | 1.66250000000e+1 | 9.01256125725e-2 | 3.60502450290e-3 |
| 3.35000000000e+0 | 1.67500000000e+1 | 8.87902330744e-2 | 3.55160932297e-3 |
| 3.37500000000e+0 | 1.68750000000e+1 | 8.74842801684e-2 | 3.49937120674e-3 |
| 3.40000000000e+0 | 1.70000000000e+1 | 8.62068965517e-2 | 3.44827586207e-3 |
| 3.42500000000e+0 | 1.71250000000e+1 | 8.49572558806e-2 | 3.39829023523e-3 |
| 3.45000000000e+0 | 1.72500000000e+1 | 8.37345614402e-2 | 3.34938245761e-3 |
| 3.47500000000e+0 | 1.73750000000e+1 | 8.25380448801e-2 | 3.30152179520e-3 |
| 3.50000000000e+0 | 1.75000000000e+1 | 8.13669650122e-2 | 3.25467860049e-3 |
| 3.52500000000e+0 | 1.76250000000e+1 | 8.02206066683e-2 | 3.20882426673e-3 |
| 3.55000000000e+0 | 1.77500000000e+1 | 7.90982796124e-2 | 3.16393118450e-3 |
| 3.57500000000e+0 | 1.78750000000e+1 | 7.79993175060e-2 | 3.11997270024e-3 |
| 3.60000000000e+0 | 1.80000000000e+1 | 7.69230769231e-2 | 3.07692307692e-3 |
| 3.62500000000e+0 | 1.81250000000e+1 | 7.58689364123e-2 | 3.03475745649e-3 |
| 3.65000000000e+0 | 1.82500000000e+1 | 7.48362956034e-2 | 2.99345182413e-3 |
| 3.67500000000e+0 | 1.83750000000e+1 | 7.38245743552e-2 | 2.95298297421e-3 |
| 3.70000000000e+0 | 1.85000000000e+1 | 7.28332119446e-2 | 2.91332847779e-3 |
| 3.72500000000e+0 | 1.86250000000e+1 | 7.18616662924e-2 | 2.87446665170e-3 |
| 3.75000000000e+0 | 1.87500000000e+1 | 7.09094132246e-2 | 2.83637652898e-3 |
| 3.77500000000e+0 | 1.88750000000e+1 | 6.99759457686e-2 | 2.79903783075e-3 |
| 3.80000000000e+0 | 1.90000000000e+1 | 6.90607734807e-2 | 2.76243093923e-3 |
| 3.82500000000e+0 | 1.91250000000e+1 | 6.81634218038e-2 | 2.72653687215e-3 |
| 3.85000000000e+0 | 1.92500000000e+1 | 6.72834314550e-2 | 2.69133725820e-3 |
| 3.87500000000e+0 | 1.93750000000e+1 | 6.64203578397e-2 | 2.65681431359e-3 |
| 3.90000000000e+0 | 1.95000000000e+1 | 6.55737704918e-2 | 2.62295081967e-3 |
| 3.92500000000e+0 | 1.96250000000e+1 | 6.47432525391e-2 | 2.58973010157e-3 |
| 3.95000000000e+0 | 1.97500000000e+1 | 6.39284001918e-2 | 2.55713600767e-3 |
| 3.97500000000e+0 | 1.98750000000e+1 | 6.31288222529e-2 | 2.52515289012e-3 |
| 4.00000000000e+0 | 2.00000000000e+1 | 6.23441396509e-2 | 2.49376558603e-3 |
| 4.02500000000e+0 | 2.01250000000e+1 | 6.15739849913e-2 | 2.46295939965e-3 |
| 4.05000000000e+0 | 2.02500000000e+1 | 6.08180021286e-2 | 2.43272008515e-3 |
| 4.07500000000e+0 | 2.03750000000e+1 | 6.00758457553e-2 | 2.40303383021e-3 |
| 4.10000000000e+0 | 2.05000000000e+1 | 5.93471810089e-2 | 2.37388724036e-3 |
| 4.12500000000e+0 | 2.06250000000e+1 | 5.86316830958e-2 | 2.34526732383e-3 |
| 4.15000000000e+0 | 2.07500000000e+1 | 5.79290369298e-2 | 2.31716147719e-3 |
| 4.17500000000e+0 | 2.08750000000e+1 | 5.72389367867e-2 | 2.28955747147e-3 |
| 4.20000000000e+0 | 2.10000000000e+1 | 5.65610859729e-2 | 2.26244343891e-3 |
| 4.22500000000e+0 | 2.11250000000e+1 | 5.58951965066e-2 | 2.23580786026e-3 |
| 4.25000000000e+0 | 2.12500000000e+1 | 5.52409888137e-2 | 2.20963955255e-3 |
| 4.27500000000e+0 | 2.13750000000e+1 | 5.45981914349e-2 | 2.18392765740e-3 |
| 4.30000000000e+0 | 2.15000000000e+1 | 5.39665407447e-2 | 2.15866162979e-3 |
| 4.32500000000e+0 | 2.16250000000e+1 | 5.33457806822e-2 | 2.13383122729e-3 |
| 4.35000000000e+0 | 2.17500000000e+1 | 5.27356624918e-2 | 2.10942649967e-3 |
| 4.37500000000e+0 | 2.18750000000e+1 | 5.21359444752e-2 | 2.08543777901e-3 |
| 4.40000000000e+0 | 2.20000000000e+1 | 5.15463917526e-2 | 2.06185567010e-3 |
| 4.42500000000e+0 | 2.21250000000e+1 | 5.09667760329e-2 | 2.03867104131e-3 |
| 4.45000000000e+0 | 2.22500000000e+1 | 5.03968753937e-2 | 2.01587501575e-3 |
| 4.47500000000e+0 | 2.23750000000e+1 | 4.98364740695e-2 | 1.99345896278e-3 |
| 4.50000000000e+0 | 2.25000000000e+1 | 4.92853622474e-2 | 1.97141448990e-3 |
| 4.52500000000e+0 | 2.26250000000e+1 | 4.87433358720e-2 | 1.94973343488e-3 |
| 4.55000000000e+0 | 2.27500000000e+1 | 4.82101964566e-2 | 1.92840785826e-3 |
| 4.57500000000e+0 | 2.28750000000e+1 | 4.76857509016e-2 | 1.90743003606e-3 |
| 4.60000000000e+0 | 2.30000000000e+1 | 4.71698113208e-2 | 1.88679245283e-3 |
| 4.62500000000e+0 | 2.31250000000e+1 | 4.66621948730e-2 | 1.86648779492e-3 |
| 4.65000000000e+0 | 2.32500000000e+1 | 4.61627236007e-2 | 1.84650894403e-3 |
| 4.67500000000e+0 | 2.33750000000e+1 | 4.56712242743e-2 | 1.82684897097e-3 |
| 4.70000000000e+0 | 2.35000000000e+1 | 4.51875282422e-2 | 1.80750112969e-3 |
| 4.72500000000e+0 | 2.36250000000e+1 | 4.47114712869e-2 | 1.78845885147e-3 |
| 4.75000000000e+0 | 2.37500000000e+1 | 4.42428934852e-2 | 1.76971573941e-3 |
| 4.77500000000e+0 | 2.38750000000e+1 | 4.37816390751e-2 | 1.75126556300e-3 |
| 4.80000000000e+0 | 2.40000000000e+1 | 4.33275563258e-2 | 1.73310225303e-3 |
| 4.82500000000e+0 | 2.41250000000e+1 | 4.28804974138e-2 | 1.71521989655e-3 |
| 4.85000000000e+0 | 2.42500000000e+1 | 4.24403183024e-2 | 1.69761273210e-3 |
| 4.87500000000e+0 | 2.43750000000e+1 | 4.20068786264e-2 | 1.68027514506e-3 |
| 4.90000000000e+0 | 2.45000000000e+1 | 4.15800415800e-2 | 1.66320166320e-3 |
| 4.92500000000e+0 | 2.46250000000e+1 | 4.11596738096e-2 | 1.64638695238e-3 |
| 4.95000000000e+0 | 2.47500000000e+1 | 4.07456453092e-2 | 1.62982581237e-3 |
| 4.97500000000e+0 | 2.48750000000e+1 | 4.03378293206e-2 | 1.61351317282e-3 |
| 5.00000000000e+0 | 2.50000000000e+1 | 3.99361022364e-2 | 1.59744408946e-3 |
Every assigned Gaussian sample for r=4/25. Rounded to 12 significant digits; kξ uses the actual ξ/a, so scaled endpoints differ between controls.
| ka | kξ | S(k)≈ | S(k)/S(0)≈ |
|---|---|---|---|
| 0 | 0 | 6.25000000000e+0 | 1.00000000000e+0 |
| 2.50000000000e-2 | 6.25000000000e-2 | 6.22568093385e+0 | 9.96108949416e-1 |
| 5.00000000000e-2 | 1.25000000000e-1 | 6.15384615385e+0 | 9.84615384615e-1 |
| 7.50000000000e-2 | 1.87500000000e-1 | 6.03773584906e+0 | 9.66037735849e-1 |
| 1.00000000000e-1 | 2.50000000000e-1 | 5.88235294118e+0 | 9.41176470588e-1 |
| 1.25000000000e-1 | 3.12500000000e-1 | 5.69395017794e+0 | 9.11032028470e-1 |
| 1.50000000000e-1 | 3.75000000000e-1 | 5.47945205479e+0 | 8.76712328767e-1 |
| 1.75000000000e-1 | 4.37500000000e-1 | 5.24590163934e+0 | 8.39344262295e-1 |
| 2.00000000000e-1 | 5.00000000000e-1 | 5.00000000000e+0 | 8.00000000000e-1 |
| 2.25000000000e-1 | 5.62500000000e-1 | 4.74777448071e+0 | 7.59643916914e-1 |
| 2.50000000000e-1 | 6.25000000000e-1 | 4.49438202247e+0 | 7.19101123596e-1 |
| 2.75000000000e-1 | 6.87500000000e-1 | 4.24403183024e+0 | 6.79045092838e-1 |
| 3.00000000000e-1 | 7.50000000000e-1 | 4.00000000000e+0 | 6.40000000000e-1 |
| 3.25000000000e-1 | 8.12500000000e-1 | 3.76470588235e+0 | 6.02352941176e-1 |
| 3.50000000000e-1 | 8.75000000000e-1 | 3.53982300885e+0 | 5.66371681416e-1 |
| 3.75000000000e-1 | 9.37500000000e-1 | 3.32640332640e+0 | 5.32224532225e-1 |
| 4.00000000000e-1 | 1.00000000000e+0 | 3.12500000000e+0 | 5.00000000000e-1 |
| 4.25000000000e-1 | 1.06250000000e+0 | 2.93577981651e+0 | 4.69724770642e-1 |
| 4.50000000000e-1 | 1.12500000000e+0 | 2.75862068966e+0 | 4.41379310345e-1 |
| 4.75000000000e-1 | 1.18750000000e+0 | 2.59319286872e+0 | 4.14910858995e-1 |
| 5.00000000000e-1 | 1.25000000000e+0 | 2.43902439024e+0 | 3.90243902439e-1 |
| 5.25000000000e-1 | 1.31250000000e+0 | 2.29555236729e+0 | 3.67288378766e-1 |
| 5.50000000000e-1 | 1.37500000000e+0 | 2.16216216216e+0 | 3.45945945946e-1 |
| 5.75000000000e-1 | 1.43750000000e+0 | 2.03821656051e+0 | 3.26114649682e-1 |
| 6.00000000000e-1 | 1.50000000000e+0 | 1.92307692308e+0 | 3.07692307692e-1 |
| 6.25000000000e-1 | 1.56250000000e+0 | 1.81611804767e+0 | 2.90578887628e-1 |
| 6.50000000000e-1 | 1.62500000000e+0 | 1.71673819742e+0 | 2.74678111588e-1 |
| 6.75000000000e-1 | 1.68750000000e+0 | 1.62436548223e+0 | 2.59898477157e-1 |
| 7.00000000000e-1 | 1.75000000000e+0 | 1.53846153846e+0 | 2.46153846154e-1 |
| 7.25000000000e-1 | 1.81250000000e+0 | 1.45852324521e+0 | 2.33363719234e-1 |
| 7.50000000000e-1 | 1.87500000000e+0 | 1.38408304498e+0 | 2.21453287197e-1 |
| 7.75000000000e-1 | 1.93750000000e+0 | 1.31470829910e+0 | 2.10353327855e-1 |
| 8.00000000000e-1 | 2.00000000000e+0 | 1.25000000000e+0 | 2.00000000000e-1 |
| 8.25000000000e-1 | 2.06250000000e+0 | 1.18959107807e+0 | 1.90334572491e-1 |
| 8.50000000000e-1 | 2.12500000000e+0 | 1.13314447592e+0 | 1.81303116147e-1 |
| 8.75000000000e-1 | 2.18750000000e+0 | 1.08035111411e+0 | 1.72856178258e-1 |
| 9.00000000000e-1 | 2.25000000000e+0 | 1.03092783505e+0 | 1.64948453608e-1 |
| 9.25000000000e-1 | 2.31250000000e+0 | 9.84615384615e-1 | 1.57538461538e-1 |
| 9.50000000000e-1 | 2.37500000000e+0 | 9.41176470588e-1 | 1.50588235294e-1 |
| 9.75000000000e-1 | 2.43750000000e+0 | 9.00393922341e-1 | 1.44063027575e-1 |
| 1.00000000000e+0 | 2.50000000000e+0 | 8.62068965517e-1 | 1.37931034483e-1 |
| 1.02500000000e+0 | 2.56250000000e+0 | 8.26019617966e-1 | 1.32163138875e-1 |
| 1.05000000000e+0 | 2.62500000000e+0 | 7.92079207921e-1 | 1.26732673267e-1 |
| 1.07500000000e+0 | 2.68750000000e+0 | 7.60095011876e-1 | 1.21615201900e-1 |
| 1.10000000000e+0 | 2.75000000000e+0 | 7.29927007299e-1 | 1.16788321168e-1 |
| 1.12500000000e+0 | 2.81250000000e+0 | 7.01446733889e-1 | 1.12231477422e-1 |
| 1.15000000000e+0 | 2.87500000000e+0 | 6.74536256324e-1 | 1.07925801012e-1 |
| 1.17500000000e+0 | 2.93750000000e+0 | 6.49087221095e-1 | 1.03853955375e-1 |
| 1.20000000000e+0 | 3.00000000000e+0 | 6.25000000000e-1 | 1.00000000000e-1 |
| 1.22500000000e+0 | 3.06250000000e+0 | 6.02182913060e-1 | 9.63492660896e-2 |
| 1.25000000000e+0 | 3.12500000000e+0 | 5.80551523948e-1 | 9.28882438316e-2 |
| 1.27500000000e+0 | 3.18750000000e+0 | 5.60028001400e-1 | 8.96044802240e-2 |
| 1.30000000000e+0 | 3.25000000000e+0 | 5.40540540541e-1 | 8.64864864865e-2 |
| 1.32500000000e+0 | 3.31250000000e+0 | 5.22022838499e-1 | 8.35236541599e-2 |
| 1.35000000000e+0 | 3.37500000000e+0 | 5.04413619168e-1 | 8.07061790668e-2 |
| 1.37500000000e+0 | 3.43750000000e+0 | 4.87656202377e-1 | 7.80249923804e-2 |
| 1.40000000000e+0 | 3.50000000000e+0 | 4.71698113208e-1 | 7.54716981132e-2 |
| 1.42500000000e+0 | 3.56250000000e+0 | 4.56490727532e-1 | 7.30385164051e-2 |
| 1.45000000000e+0 | 3.62500000000e+0 | 4.41988950276e-1 | 7.07182320442e-2 |
| 1.47500000000e+0 | 3.68750000000e+0 | 4.28150923200e-1 | 6.85041477121e-2 |
| 1.50000000000e+0 | 3.75000000000e+0 | 4.14937759336e-1 | 6.63900414938e-2 |
| 1.52500000000e+0 | 3.81250000000e+0 | 4.02313301484e-1 | 6.43701282374e-2 |
| 1.55000000000e+0 | 3.87500000000e+0 | 3.90243902439e-1 | 6.24390243902e-2 |
| 1.57500000000e+0 | 3.93750000000e+0 | 3.78698224852e-1 | 6.05917159763e-2 |
| 1.60000000000e+0 | 4.00000000000e+0 | 3.67647058824e-1 | 5.88235294118e-2 |
| 1.62500000000e+0 | 4.06250000000e+0 | 3.57063155546e-1 | 5.71301048873e-2 |
| 1.65000000000e+0 | 4.12500000000e+0 | 3.46921075455e-1 | 5.55073720729e-2 |
| 1.67500000000e+0 | 4.18750000000e+0 | 3.37197049526e-1 | 5.39515279241e-2 |
| 1.70000000000e+0 | 4.25000000000e+0 | 3.27868852459e-1 | 5.24590163934e-2 |
| 1.72500000000e+0 | 4.31250000000e+0 | 3.18915686665e-1 | 5.10265098665e-2 |
| 1.75000000000e+0 | 4.37500000000e+0 | 3.10318076028e-1 | 4.96508921645e-2 |
| 1.77500000000e+0 | 4.43750000000e+0 | 3.02057768548e-1 | 4.83292429677e-2 |
| 1.80000000000e+0 | 4.50000000000e+0 | 2.94117647059e-1 | 4.70588235294e-2 |
| 1.82500000000e+0 | 4.56250000000e+0 | 2.86481647269e-1 | 4.58370635631e-2 |
| 1.85000000000e+0 | 4.62500000000e+0 | 2.79134682484e-1 | 4.46615491975e-2 |
| 1.87500000000e+0 | 4.68750000000e+0 | 2.72062574392e-1 | 4.35300119027e-2 |
| 1.90000000000e+0 | 4.75000000000e+0 | 2.65251989390e-1 | 4.24403183024e-2 |
| 1.92500000000e+0 | 4.81250000000e+0 | 2.58690379951e-1 | 4.13904607922e-2 |
| 1.95000000000e+0 | 4.87500000000e+0 | 2.52365930599e-1 | 4.03785488959e-2 |
| 1.97500000000e+0 | 4.93750000000e+0 | 2.46267508081e-1 | 3.94028012929e-2 |
| 2.00000000000e+0 | 5.00000000000e+0 | 2.40384615385e-1 | 3.84615384615e-2 |
| 2.02500000000e+0 | 5.06250000000e+0 | 2.34707349274e-1 | 3.75531758838e-2 |
| 2.05000000000e+0 | 5.12500000000e+0 | 2.29226361032e-1 | 3.66762177650e-2 |
| 2.07500000000e+0 | 5.18750000000e+0 | 2.23932820154e-1 | 3.58292512246e-2 |
| 2.10000000000e+0 | 5.25000000000e+0 | 2.18818380744e-1 | 3.50109409190e-2 |
| 2.12500000000e+0 | 5.31250000000e+0 | 2.13875150381e-1 | 3.42200240610e-2 |
| 2.15000000000e+0 | 5.37500000000e+0 | 2.09095661265e-1 | 3.34553058024e-2 |
| 2.17500000000e+0 | 5.43750000000e+0 | 2.04472843450e-1 | 3.27156549521e-2 |
| 2.20000000000e+0 | 5.50000000000e+0 | 2.00000000000e-1 | 3.20000000000e-2 |
| 2.22500000000e+0 | 5.56250000000e+0 | 1.95670783906e-1 | 3.13073254250e-2 |
| 2.25000000000e+0 | 5.62500000000e+0 | 1.91479176640e-1 | 3.06366682623e-2 |
| 2.27500000000e+0 | 5.68750000000e+0 | 1.87419468197e-1 | 2.99871149116e-2 |
| 2.30000000000e+0 | 5.75000000000e+0 | 1.83486238532e-1 | 2.93577981651e-2 |
| 2.32500000000e+0 | 5.81250000000e+0 | 1.79674340258e-1 | 2.87478944413e-2 |
| 2.35000000000e+0 | 5.87500000000e+0 | 1.75978882534e-1 | 2.81566212055e-2 |
| 2.37500000000e+0 | 5.93750000000e+0 | 1.72395216033e-1 | 2.75832345652e-2 |
| 2.40000000000e+0 | 6.00000000000e+0 | 1.68918918919e-1 | 2.70270270270e-2 |
| 2.42500000000e+0 | 6.06250000000e+0 | 1.65545783756e-1 | 2.64873254009e-2 |
| 2.45000000000e+0 | 6.12500000000e+0 | 1.62271805274e-1 | 2.59634888438e-2 |
| 2.47500000000e+0 | 6.18750000000e+0 | 1.59093168937e-1 | 2.54549070299e-2 |
| 2.50000000000e+0 | 6.25000000000e+0 | 1.56006240250e-1 | 2.49609984399e-2 |
| 2.52500000000e+0 | 6.31250000000e+0 | 1.53007554748e-1 | 2.44812087597e-2 |
| 2.55000000000e+0 | 6.37500000000e+0 | 1.50093808630e-1 | 2.40150093809e-2 |
| 2.57500000000e+0 | 6.43750000000e+0 | 1.47261849977e-1 | 2.35618959963e-2 |
| 2.60000000000e+0 | 6.50000000000e+0 | 1.44508670520e-1 | 2.31213872832e-2 |
| 2.62500000000e+0 | 6.56250000000e+0 | 1.41831397926e-1 | 2.26930236681e-2 |
| 2.65000000000e+0 | 6.62500000000e+0 | 1.39227288549e-1 | 2.22763661678e-2 |
| 2.67500000000e+0 | 6.68750000000e+0 | 1.36693720632e-1 | 2.18709953012e-2 |
| 2.70000000000e+0 | 6.75000000000e+0 | 1.34228187919e-1 | 2.14765100671e-2 |
| 2.72500000000e+0 | 6.81250000000e+0 | 1.31828293648e-1 | 2.10925269836e-2 |
| 2.75000000000e+0 | 6.87500000000e+0 | 1.29491744901e-1 | 2.07186791842e-2 |
| 2.77500000000e+0 | 6.93750000000e+0 | 1.27216347301e-1 | 2.03546155681e-2 |
| 2.80000000000e+0 | 7.00000000000e+0 | 1.25000000000e-1 | 2.00000000000e-2 |
| 2.82500000000e+0 | 7.06250000000e+0 | 1.22840690979e-1 | 1.96545105566e-2 |
| 2.85000000000e+0 | 7.12500000000e+0 | 1.20736492605e-1 | 1.93178388168e-2 |
| 2.87500000000e+0 | 7.18750000000e+0 | 1.18685557451e-1 | 1.89896891922e-2 |
| 2.90000000000e+0 | 7.25000000000e+0 | 1.16686114352e-1 | 1.86697782964e-2 |
| 2.92500000000e+0 | 7.31250000000e+0 | 1.14736464683e-1 | 1.83578343492e-2 |
| 2.95000000000e+0 | 7.37500000000e+0 | 1.12834978843e-1 | 1.80535966150e-2 |
| 2.97500000000e+0 | 7.43750000000e+0 | 1.10980092946e-1 | 1.77568148713e-2 |
| 3.00000000000e+0 | 7.50000000000e+0 | 1.09170305677e-1 | 1.74672489083e-2 |
| 3.02500000000e+0 | 7.56250000000e+0 | 1.07404175337e-1 | 1.71846680540e-2 |
| 3.05000000000e+0 | 7.62500000000e+0 | 1.05680317041e-1 | 1.69088507266e-2 |
| 3.07500000000e+0 | 7.68750000000e+0 | 1.03997400065e-1 | 1.66395840104e-2 |
| 3.10000000000e+0 | 7.75000000000e+0 | 1.02354145343e-1 | 1.63766632549e-2 |
| 3.12500000000e+0 | 7.81250000000e+0 | 1.00749323090e-1 | 1.61198916945e-2 |
| 3.15000000000e+0 | 7.87500000000e+0 | 9.91817505579e-2 | 1.58690800893e-2 |
| 3.17500000000e+0 | 7.93750000000e+0 | 9.76502898993e-2 | 1.56240463839e-2 |
| 3.20000000000e+0 | 8.00000000000e+0 | 9.61538461538e-2 | 1.53846153846e-2 |
| 3.22500000000e+0 | 8.06250000000e+0 | 9.46913653311e-2 | 1.51506184530e-2 |
| 3.25000000000e+0 | 8.12500000000e+0 | 9.32618325950e-2 | 1.49218932152e-2 |
| 3.27500000000e+0 | 8.18750000000e+0 | 9.18642705403e-2 | 1.46982832864e-2 |
| 3.30000000000e+0 | 8.25000000000e+0 | 9.04977375566e-2 | 1.44796380090e-2 |
| 3.32500000000e+0 | 8.31250000000e+0 | 8.91613262747e-2 | 1.42658122040e-2 |
| 3.35000000000e+0 | 8.37500000000e+0 | 8.78541620909e-2 | 1.40566659345e-2 |
| 3.37500000000e+0 | 8.43750000000e+0 | 8.65754017640e-2 | 1.38520642822e-2 |
| 3.40000000000e+0 | 8.50000000000e+0 | 8.53242320819e-2 | 1.36518771331e-2 |
| 3.42500000000e+0 | 8.56250000000e+0 | 8.40998685940e-2 | 1.34559789750e-2 |
| 3.45000000000e+0 | 8.62500000000e+0 | 8.29015544041e-2 | 1.32642487047e-2 |
| 3.47500000000e+0 | 8.68750000000e+0 | 8.17285590233e-2 | 1.30765694437e-2 |
| 3.50000000000e+0 | 8.75000000000e+0 | 8.05801772764e-2 | 1.28928283642e-2 |
| 3.52500000000e+0 | 8.81250000000e+0 | 7.94557282614e-2 | 1.27129165218e-2 |
| 3.55000000000e+0 | 8.87500000000e+0 | 7.83545543585e-2 | 1.25367286974e-2 |
| 3.57500000000e+0 | 8.93750000000e+0 | 7.72760202850e-2 | 1.23641632456e-2 |
| 3.60000000000e+0 | 9.00000000000e+0 | 7.62195121951e-2 | 1.21951219512e-2 |
| 3.62500000000e+0 | 9.06250000000e+0 | 7.51844368216e-2 | 1.20295098915e-2 |
| 3.65000000000e+0 | 9.12500000000e+0 | 7.41702206564e-2 | 1.18672353050e-2 |
| 3.67500000000e+0 | 9.18750000000e+0 | 7.31763091699e-2 | 1.17082094672e-2 |
| 3.70000000000e+0 | 9.25000000000e+0 | 7.22021660650e-2 | 1.15523465704e-2 |
| 3.72500000000e+0 | 9.31250000000e+0 | 7.12472725653e-2 | 1.13995636105e-2 |
| 3.75000000000e+0 | 9.37500000000e+0 | 7.03111267358e-2 | 1.12497802777e-2 |
| 3.77500000000e+0 | 9.43750000000e+0 | 6.93932428330e-2 | 1.11029188533e-2 |
| 3.80000000000e+0 | 9.50000000000e+0 | 6.84931506849e-2 | 1.09589041096e-2 |
| 3.82500000000e+0 | 9.56250000000e+0 | 6.76103950982e-2 | 1.08176632157e-2 |
| 3.85000000000e+0 | 9.62500000000e+0 | 6.67445352912e-2 | 1.06791256466e-2 |
| 3.87500000000e+0 | 9.68750000000e+0 | 6.58951443516e-2 | 1.05432230962e-2 |
| 3.90000000000e+0 | 9.75000000000e+0 | 6.50618087183e-2 | 1.04098893949e-2 |
| 3.92500000000e+0 | 9.81250000000e+0 | 6.42441276852e-2 | 1.02790604296e-2 |
| 3.95000000000e+0 | 9.87500000000e+0 | 6.34417129262e-2 | 1.01506740682e-2 |
| 3.97500000000e+0 | 9.93750000000e+0 | 6.26541880409e-2 | 1.00246700865e-2 |
| 4.00000000000e+0 | 1.00000000000e+1 | 6.18811881188e-2 | 9.90099009901e-3 |
| 4.02500000000e+0 | 1.00625000000e+1 | 6.11223593231e-2 | 9.77957749169e-3 |
| 4.05000000000e+0 | 1.01250000000e+1 | 6.03773584906e-2 | 9.66037735849e-3 |
| 4.07500000000e+0 | 1.01875000000e+1 | 5.96458527493e-2 | 9.54333643989e-3 |
| 4.10000000000e+0 | 1.02500000000e+1 | 5.89275191514e-2 | 9.42840306423e-3 |
| 4.12500000000e+0 | 1.03125000000e+1 | 5.82220443215e-2 | 9.31552709144e-3 |
| 4.15000000000e+0 | 1.03750000000e+1 | 5.75291241191e-2 | 9.20465985905e-3 |
| 4.17500000000e+0 | 1.04375000000e+1 | 5.68484633150e-2 | 9.09575413040e-3 |
| 4.20000000000e+0 | 1.05000000000e+1 | 5.61797752809e-2 | 8.98876404494e-3 |
| 4.22500000000e+0 | 1.05625000000e+1 | 5.55227816914e-2 | 8.88364507062e-3 |
| 4.25000000000e+0 | 1.06250000000e+1 | 5.48772122376e-2 | 8.78035395802e-3 |
| 4.27500000000e+0 | 1.06875000000e+1 | 5.42428043530e-2 | 8.67884869648e-3 |
| 4.30000000000e+0 | 1.07500000000e+1 | 5.36193029491e-2 | 8.57908847185e-3 |
| 4.32500000000e+0 | 1.08125000000e+1 | 5.30064601623e-2 | 8.48103362597e-3 |
| 4.35000000000e+0 | 1.08750000000e+1 | 5.24040351107e-2 | 8.38464561771e-3 |
| 4.37500000000e+0 | 1.09375000000e+1 | 5.18117936595e-2 | 8.28988698553e-3 |
| 4.40000000000e+0 | 1.10000000000e+1 | 5.12295081967e-2 | 8.19672131148e-3 |
| 4.42500000000e+0 | 1.10625000000e+1 | 5.06569574165e-2 | 8.10511318664e-3 |
| 4.45000000000e+0 | 1.11250000000e+1 | 5.00939261115e-2 | 8.01502817783e-3 |
| 4.47500000000e+0 | 1.11875000000e+1 | 4.95402049726e-2 | 7.92643279562e-3 |
| 4.50000000000e+0 | 1.12500000000e+1 | 4.89955903969e-2 | 7.83929446350e-3 |
| 4.52500000000e+0 | 1.13125000000e+1 | 4.84598843020e-2 | 7.75358148832e-3 |
| 4.55000000000e+0 | 1.13750000000e+1 | 4.79328939485e-2 | 7.66926303176e-3 |
| 4.57500000000e+0 | 1.14375000000e+1 | 4.74144317677e-2 | 7.58630908283e-3 |
| 4.60000000000e+0 | 1.15000000000e+1 | 4.69043151970e-2 | 7.50469043152e-3 |
| 4.62500000000e+0 | 1.15625000000e+1 | 4.64023665207e-2 | 7.42437864331e-3 |
| 4.65000000000e+0 | 1.16250000000e+1 | 4.59084127166e-2 | 7.34534603466e-3 |
| 4.67500000000e+0 | 1.16875000000e+1 | 4.54222853087e-2 | 7.26756564940e-3 |
| 4.70000000000e+0 | 1.17500000000e+1 | 4.49438202247e-2 | 7.19101123596e-3 |
| 4.72500000000e+0 | 1.18125000000e+1 | 4.44728576591e-2 | 7.11565722545e-3 |
| 4.75000000000e+0 | 1.18750000000e+1 | 4.40092419408e-2 | 7.04147871053e-3 |
| 4.77500000000e+0 | 1.19375000000e+1 | 4.35528214062e-2 | 6.96845142499e-3 |
| 4.80000000000e+0 | 1.20000000000e+1 | 4.31034482759e-2 | 6.89655172414e-3 |
| 4.82500000000e+0 | 1.20625000000e+1 | 4.26609785362e-2 | 6.82575656579e-3 |
| 4.85000000000e+0 | 1.21250000000e+1 | 4.22252718252e-2 | 6.75604349203e-3 |
| 4.87500000000e+0 | 1.21875000000e+1 | 4.17961913221e-2 | 6.68739061153e-3 |
| 4.90000000000e+0 | 1.22500000000e+1 | 4.13736036409e-2 | 6.61977658254e-3 |
| 4.92500000000e+0 | 1.23125000000e+1 | 4.09573787278e-2 | 6.55318059644e-3 |
| 4.95000000000e+0 | 1.23750000000e+1 | 4.05473897618e-2 | 6.48758236189e-3 |
| 4.97500000000e+0 | 1.24375000000e+1 | 4.01435130592e-2 | 6.42296208947e-3 |
| 5.00000000000e+0 | 1.25000000000e+1 | 3.97456279809e-2 | 6.35930047695e-3 |
Every assigned Gaussian sample for r=1. Rounded to 12 significant digits; kξ uses the actual ξ/a, so scaled endpoints differ between controls.
| ka | kξ | S(k)≈ | S(k)/S(0)≈ |
|---|---|---|---|
| 0 | 0 | 1.00000000000e+0 | 1.00000000000e+0 |
| 2.50000000000e-2 | 2.50000000000e-2 | 9.99375390381e-1 | 9.99375390381e-1 |
| 5.00000000000e-2 | 5.00000000000e-2 | 9.97506234414e-1 | 9.97506234414e-1 |
| 7.50000000000e-2 | 7.50000000000e-2 | 9.94406463642e-1 | 9.94406463642e-1 |
| 1.00000000000e-1 | 1.00000000000e-1 | 9.90099009901e-1 | 9.90099009901e-1 |
| 1.25000000000e-1 | 1.25000000000e-1 | 9.84615384615e-1 | 9.84615384615e-1 |
| 1.50000000000e-1 | 1.50000000000e-1 | 9.77995110024e-1 | 9.77995110024e-1 |
| 1.75000000000e-1 | 1.75000000000e-1 | 9.70285021225e-1 | 9.70285021225e-1 |
| 2.00000000000e-1 | 2.00000000000e-1 | 9.61538461538e-1 | 9.61538461538e-1 |
| 2.25000000000e-1 | 2.25000000000e-1 | 9.51814396193e-1 | 9.51814396193e-1 |
| 2.50000000000e-1 | 2.50000000000e-1 | 9.41176470588e-1 | 9.41176470588e-1 |
| 2.75000000000e-1 | 2.75000000000e-1 | 9.29692039512e-1 | 9.29692039512e-1 |
| 3.00000000000e-1 | 3.00000000000e-1 | 9.17431192661e-1 | 9.17431192661e-1 |
| 3.25000000000e-1 | 3.25000000000e-1 | 9.04465799887e-1 | 9.04465799887e-1 |
| 3.50000000000e-1 | 3.50000000000e-1 | 8.90868596882e-1 | 8.90868596882e-1 |
| 3.75000000000e-1 | 3.75000000000e-1 | 8.76712328767e-1 | 8.76712328767e-1 |
| 4.00000000000e-1 | 4.00000000000e-1 | 8.62068965517e-1 | 8.62068965517e-1 |
| 4.25000000000e-1 | 4.25000000000e-1 | 8.47008999471e-1 | 8.47008999471e-1 |
| 4.50000000000e-1 | 4.50000000000e-1 | 8.31600831601e-1 | 8.31600831601e-1 |
| 4.75000000000e-1 | 4.75000000000e-1 | 8.15910249873e-1 | 8.15910249873e-1 |
| 5.00000000000e-1 | 5.00000000000e-1 | 8.00000000000e-1 | 8.00000000000e-1 |
| 5.25000000000e-1 | 5.25000000000e-1 | 7.83929446350e-1 | 7.83929446350e-1 |
| 5.50000000000e-1 | 5.50000000000e-1 | 7.67754318618e-1 | 7.67754318618e-1 |
| 5.75000000000e-1 | 5.75000000000e-1 | 7.51526538281e-1 | 7.51526538281e-1 |
| 6.00000000000e-1 | 6.00000000000e-1 | 7.35294117647e-1 | 7.35294117647e-1 |
| 6.25000000000e-1 | 6.25000000000e-1 | 7.19101123596e-1 | 7.19101123596e-1 |
| 6.50000000000e-1 | 6.50000000000e-1 | 7.02987697715e-1 | 7.02987697715e-1 |
| 6.75000000000e-1 | 6.75000000000e-1 | 6.86990124517e-1 | 6.86990124517e-1 |
| 7.00000000000e-1 | 7.00000000000e-1 | 6.71140939597e-1 | 6.71140939597e-1 |
| 7.25000000000e-1 | 7.25000000000e-1 | 6.55469070053e-1 | 6.55469070053e-1 |
| 7.50000000000e-1 | 7.50000000000e-1 | 6.40000000000e-1 | 6.40000000000e-1 |
| 7.75000000000e-1 | 7.75000000000e-1 | 6.24755954705e-1 | 6.24755954705e-1 |
| 8.00000000000e-1 | 8.00000000000e-1 | 6.09756097561e-1 | 6.09756097561e-1 |
| 8.25000000000e-1 | 8.25000000000e-1 | 5.95016734846e-1 | 5.95016734846e-1 |
| 8.50000000000e-1 | 8.50000000000e-1 | 5.80551523948e-1 | 5.80551523948e-1 |
| 8.75000000000e-1 | 8.75000000000e-1 | 5.66371681416e-1 | 5.66371681416e-1 |
| 9.00000000000e-1 | 9.00000000000e-1 | 5.52486187845e-1 | 5.52486187845e-1 |
| 9.25000000000e-1 | 9.25000000000e-1 | 5.38901987201e-1 | 5.38901987201e-1 |
| 9.50000000000e-1 | 9.50000000000e-1 | 5.25624178712e-1 | 5.25624178712e-1 |
| 9.75000000000e-1 | 9.75000000000e-1 | 5.12656199936e-1 | 5.12656199936e-1 |
| 1.00000000000e+0 | 1.00000000000e+0 | 5.00000000000e-1 | 5.00000000000e-1 |
| 1.02500000000e+0 | 1.02500000000e+0 | 4.87656202377e-1 | 4.87656202377e-1 |
| 1.05000000000e+0 | 1.05000000000e+0 | 4.75624256837e-1 | 4.75624256837e-1 |
| 1.07500000000e+0 | 1.07500000000e+0 | 4.63902580458e-1 | 4.63902580458e-1 |
| 1.10000000000e+0 | 1.10000000000e+0 | 4.52488687783e-1 | 4.52488687783e-1 |
| 1.12500000000e+0 | 1.12500000000e+0 | 4.41379310345e-1 | 4.41379310345e-1 |
| 1.15000000000e+0 | 1.15000000000e+0 | 4.30570505920e-1 | 4.30570505920e-1 |
| 1.17500000000e+0 | 1.17500000000e+0 | 4.20057757942e-1 | 4.20057757942e-1 |
| 1.20000000000e+0 | 1.20000000000e+0 | 4.09836065574e-1 | 4.09836065574e-1 |
| 1.22500000000e+0 | 1.22500000000e+0 | 3.99900024994e-1 | 3.99900024994e-1 |
| 1.25000000000e+0 | 1.25000000000e+0 | 3.90243902439e-1 | 3.90243902439e-1 |
| 1.27500000000e+0 | 1.27500000000e+0 | 3.80861699595e-1 | 3.80861699595e-1 |
| 1.30000000000e+0 | 1.30000000000e+0 | 3.71747211896e-1 | 3.71747211896e-1 |
| 1.32500000000e+0 | 1.32500000000e+0 | 3.62894080290e-1 | 3.62894080290e-1 |
| 1.35000000000e+0 | 1.35000000000e+0 | 3.54295837024e-1 | 3.54295837024e-1 |
| 1.37500000000e+0 | 1.37500000000e+0 | 3.45945945946e-1 | 3.45945945946e-1 |
| 1.40000000000e+0 | 1.40000000000e+0 | 3.37837837838e-1 | 3.37837837838e-1 |
| 1.42500000000e+0 | 1.42500000000e+0 | 3.29964941225e-1 | 3.29964941225e-1 |
| 1.45000000000e+0 | 1.45000000000e+0 | 3.22320709106e-1 | 3.22320709106e-1 |
| 1.47500000000e+0 | 1.47500000000e+0 | 3.14898642000e-1 | 3.14898642000e-1 |
| 1.50000000000e+0 | 1.50000000000e+0 | 3.07692307692e-1 | 3.07692307692e-1 |
| 1.52500000000e+0 | 1.52500000000e+0 | 3.00695358015e-1 | 3.00695358015e-1 |
| 1.55000000000e+0 | 1.55000000000e+0 | 2.93901542983e-1 | 2.93901542983e-1 |
| 1.57500000000e+0 | 1.57500000000e+0 | 2.87304722571e-1 | 2.87304722571e-1 |
| 1.60000000000e+0 | 1.60000000000e+0 | 2.80898876404e-1 | 2.80898876404e-1 |
| 1.62500000000e+0 | 1.62500000000e+0 | 2.74678111588e-1 | 2.74678111588e-1 |
| 1.65000000000e+0 | 1.65000000000e+0 | 2.68636668905e-1 | 2.68636668905e-1 |
| 1.67500000000e+0 | 1.67500000000e+0 | 2.62768927574e-1 | 2.62768927574e-1 |
| 1.70000000000e+0 | 1.70000000000e+0 | 2.57069408740e-1 | 2.57069408740e-1 |
| 1.72500000000e+0 | 1.72500000000e+0 | 2.51532777865e-1 | 2.51532777865e-1 |
| 1.75000000000e+0 | 1.75000000000e+0 | 2.46153846154e-1 | 2.46153846154e-1 |
| 1.77500000000e+0 | 1.77500000000e+0 | 2.40927571149e-1 | 2.40927571149e-1 |
| 1.80000000000e+0 | 1.80000000000e+0 | 2.35849056604e-1 | 2.35849056604e-1 |
| 1.82500000000e+0 | 1.82500000000e+0 | 2.30913551739e-1 | 2.30913551739e-1 |
| 1.85000000000e+0 | 1.85000000000e+0 | 2.26116449972e-1 | 2.26116449972e-1 |
| 1.87500000000e+0 | 1.87500000000e+0 | 2.21453287197e-1 | 2.21453287197e-1 |
| 1.90000000000e+0 | 1.90000000000e+0 | 2.16919739696e-1 | 2.16919739696e-1 |
| 1.92500000000e+0 | 1.92500000000e+0 | 2.12511621729e-1 | 2.12511621729e-1 |
| 1.95000000000e+0 | 1.95000000000e+0 | 2.08224882874e-1 | 2.08224882874e-1 |
| 1.97500000000e+0 | 1.97500000000e+0 | 2.04055605152e-1 | 2.04055605152e-1 |
| 2.00000000000e+0 | 2.00000000000e+0 | 2.00000000000e-1 | 2.00000000000e-1 |
| 2.02500000000e+0 | 2.02500000000e+0 | 1.96054405097e-1 | 1.96054405097e-1 |
| 2.05000000000e+0 | 2.05000000000e+0 | 1.92215281115e-1 | 1.92215281115e-1 |
| 2.07500000000e+0 | 2.07500000000e+0 | 1.88479208387e-1 | 1.88479208387e-1 |
| 2.10000000000e+0 | 2.10000000000e+0 | 1.84842883549e-1 | 1.84842883549e-1 |
| 2.12500000000e+0 | 2.12500000000e+0 | 1.81303116147e-1 | 1.81303116147e-1 |
| 2.15000000000e+0 | 2.15000000000e+0 | 1.77856825256e-1 | 1.77856825256e-1 |
| 2.17500000000e+0 | 2.17500000000e+0 | 1.74501036100e-1 | 1.74501036100e-1 |
| 2.20000000000e+0 | 2.20000000000e+0 | 1.71232876712e-1 | 1.71232876712e-1 |
| 2.22500000000e+0 | 2.22500000000e+0 | 1.68049574625e-1 | 1.68049574625e-1 |
| 2.25000000000e+0 | 2.25000000000e+0 | 1.64948453608e-1 | 1.64948453608e-1 |
| 2.27500000000e+0 | 2.27500000000e+0 | 1.61926930473e-1 | 1.61926930473e-1 |
| 2.30000000000e+0 | 2.30000000000e+0 | 1.58982511924e-1 | 1.58982511924e-1 |
| 2.32500000000e+0 | 2.32500000000e+0 | 1.56112791492e-1 | 1.56112791492e-1 |
| 2.35000000000e+0 | 2.35000000000e+0 | 1.53315446531e-1 | 1.53315446531e-1 |
| 2.37500000000e+0 | 2.37500000000e+0 | 1.50588235294e-1 | 1.50588235294e-1 |
| 2.40000000000e+0 | 2.40000000000e+0 | 1.47928994083e-1 | 1.47928994083e-1 |
| 2.42500000000e+0 | 2.42500000000e+0 | 1.45335634481e-1 | 1.45335634481e-1 |
| 2.45000000000e+0 | 2.45000000000e+0 | 1.42806140664e-1 | 1.42806140664e-1 |
| 2.47500000000e+0 | 2.47500000000e+0 | 1.40338566792e-1 | 1.40338566792e-1 |
| 2.50000000000e+0 | 2.50000000000e+0 | 1.37931034483e-1 | 1.37931034483e-1 |
| 2.52500000000e+0 | 2.52500000000e+0 | 1.35581730362e-1 | 1.35581730362e-1 |
| 2.55000000000e+0 | 2.55000000000e+0 | 1.33288903699e-1 | 1.33288903699e-1 |
| 2.57500000000e+0 | 2.57500000000e+0 | 1.31050864117e-1 | 1.31050864117e-1 |
| 2.60000000000e+0 | 2.60000000000e+0 | 1.28865979381e-1 | 1.28865979381e-1 |
| 2.62500000000e+0 | 2.62500000000e+0 | 1.26732673267e-1 | 1.26732673267e-1 |
| 2.65000000000e+0 | 2.65000000000e+0 | 1.24649423496e-1 | 1.24649423496e-1 |
| 2.67500000000e+0 | 2.67500000000e+0 | 1.22614759752e-1 | 1.22614759752e-1 |
| 2.70000000000e+0 | 2.70000000000e+0 | 1.20627261761e-1 | 1.20627261761e-1 |
| 2.72500000000e+0 | 2.72500000000e+0 | 1.18685557451e-1 | 1.18685557451e-1 |
| 2.75000000000e+0 | 2.75000000000e+0 | 1.16788321168e-1 | 1.16788321168e-1 |
| 2.77500000000e+0 | 2.77500000000e+0 | 1.14934271963e-1 | 1.14934271963e-1 |
| 2.80000000000e+0 | 2.80000000000e+0 | 1.13122171946e-1 | 1.13122171946e-1 |
| 2.82500000000e+0 | 2.82500000000e+0 | 1.11350824692e-1 | 1.11350824692e-1 |
| 2.85000000000e+0 | 2.85000000000e+0 | 1.09619073719e-1 | 1.09619073719e-1 |
| 2.87500000000e+0 | 2.87500000000e+0 | 1.07925801012e-1 | 1.07925801012e-1 |
| 2.90000000000e+0 | 2.90000000000e+0 | 1.06269925611e-1 | 1.06269925611e-1 |
| 2.92500000000e+0 | 2.92500000000e+0 | 1.04650402250e-1 | 1.04650402250e-1 |
| 2.95000000000e+0 | 2.95000000000e+0 | 1.03066220046e-1 | 1.03066220046e-1 |
| 2.97500000000e+0 | 2.97500000000e+0 | 1.01516401244e-1 | 1.01516401244e-1 |
| 3.00000000000e+0 | 3.00000000000e+0 | 1.00000000000e-1 | 1.00000000000e-1 |
| 3.02500000000e+0 | 3.02500000000e+0 | 9.85161012253e-2 | 9.85161012253e-2 |
| 3.05000000000e+0 | 3.05000000000e+0 | 9.70638194613e-2 | 9.70638194613e-2 |
| 3.07500000000e+0 | 3.07500000000e+0 | 9.56422978062e-2 | 9.56422978062e-2 |
| 3.10000000000e+0 | 3.10000000000e+0 | 9.42507068803e-2 | 9.42507068803e-2 |
| 3.12500000000e+0 | 3.12500000000e+0 | 9.28882438316e-2 | 9.28882438316e-2 |
| 3.15000000000e+0 | 3.15000000000e+0 | 9.15541313802e-2 | 9.15541313802e-2 |
| 3.17500000000e+0 | 3.17500000000e+0 | 9.02476168989e-2 | 9.02476168989e-2 |
| 3.20000000000e+0 | 3.20000000000e+0 | 8.89679715302e-2 | 8.89679715302e-2 |
| 3.22500000000e+0 | 3.22500000000e+0 | 8.77144893372e-2 | 8.77144893372e-2 |
| 3.25000000000e+0 | 3.25000000000e+0 | 8.64864864865e-2 | 8.64864864865e-2 |
| 3.27500000000e+0 | 3.27500000000e+0 | 8.52833004637e-2 | 8.52833004637e-2 |
| 3.30000000000e+0 | 3.30000000000e+0 | 8.41042893188e-2 | 8.41042893188e-2 |
| 3.32500000000e+0 | 3.32500000000e+0 | 8.29488309399e-2 | 8.29488309399e-2 |
| 3.35000000000e+0 | 3.35000000000e+0 | 8.18163223563e-2 | 8.18163223563e-2 |
| 3.37500000000e+0 | 3.37500000000e+0 | 8.07061790668e-2 | 8.07061790668e-2 |
| 3.40000000000e+0 | 3.40000000000e+0 | 7.96178343949e-2 | 7.96178343949e-2 |
| 3.42500000000e+0 | 3.42500000000e+0 | 7.85507388679e-2 | 7.85507388679e-2 |
| 3.45000000000e+0 | 3.45000000000e+0 | 7.75043596202e-2 | 7.75043596202e-2 |
| 3.47500000000e+0 | 3.47500000000e+0 | 7.64781798193e-2 | 7.64781798193e-2 |
| 3.50000000000e+0 | 3.50000000000e+0 | 7.54716981132e-2 | 7.54716981132e-2 |
| 3.52500000000e+0 | 3.52500000000e+0 | 7.44844280993e-2 | 7.44844280993e-2 |
| 3.55000000000e+0 | 3.55000000000e+0 | 7.35158978129e-2 | 7.35158978129e-2 |
| 3.57500000000e+0 | 3.57500000000e+0 | 7.25656492358e-2 | 7.25656492358e-2 |
| 3.60000000000e+0 | 3.60000000000e+0 | 7.16332378223e-2 | 7.16332378223e-2 |
| 3.62500000000e+0 | 3.62500000000e+0 | 7.07182320442e-2 | 7.07182320442e-2 |
| 3.65000000000e+0 | 3.65000000000e+0 | 6.98202129516e-2 | 6.98202129516e-2 |
| 3.67500000000e+0 | 3.67500000000e+0 | 6.89387737516e-2 | 6.89387737516e-2 |
| 3.70000000000e+0 | 3.70000000000e+0 | 6.80735194010e-2 | 6.80735194010e-2 |
| 3.72500000000e+0 | 3.72500000000e+0 | 6.72240662157e-2 | 6.72240662157e-2 |
| 3.75000000000e+0 | 3.75000000000e+0 | 6.63900414938e-2 | 6.63900414938e-2 |
| 3.77500000000e+0 | 3.77500000000e+0 | 6.55710831523e-2 | 6.55710831523e-2 |
| 3.80000000000e+0 | 3.80000000000e+0 | 6.47668393782e-2 | 6.47668393782e-2 |
| 3.82500000000e+0 | 3.82500000000e+0 | 6.39769682914e-2 | 6.39769682914e-2 |
| 3.85000000000e+0 | 3.85000000000e+0 | 6.32011376205e-2 | 6.32011376205e-2 |
| 3.87500000000e+0 | 3.87500000000e+0 | 6.24390243902e-2 | 6.24390243902e-2 |
| 3.90000000000e+0 | 3.90000000000e+0 | 6.16903146206e-2 | 6.16903146206e-2 |
| 3.92500000000e+0 | 3.92500000000e+0 | 6.09547030363e-2 | 6.09547030363e-2 |
| 3.95000000000e+0 | 3.95000000000e+0 | 6.02318927872e-2 | 6.02318927872e-2 |
| 3.97500000000e+0 | 3.97500000000e+0 | 5.95215951788e-2 | 5.95215951788e-2 |
| 4.00000000000e+0 | 4.00000000000e+0 | 5.88235294118e-2 | 5.88235294118e-2 |
| 4.02500000000e+0 | 4.02500000000e+0 | 5.81374223320e-2 | 5.81374223320e-2 |
| 4.05000000000e+0 | 4.05000000000e+0 | 5.74630081885e-2 | 5.74630081885e-2 |
| 4.07500000000e+0 | 4.07500000000e+0 | 5.68000284000e-2 | 5.68000284000e-2 |
| 4.10000000000e+0 | 4.10000000000e+0 | 5.61482313307e-2 | 5.61482313307e-2 |
| 4.12500000000e+0 | 4.12500000000e+0 | 5.55073720729e-2 | 5.55073720729e-2 |
| 4.15000000000e+0 | 4.15000000000e+0 | 5.48772122376e-2 | 5.48772122376e-2 |
| 4.17500000000e+0 | 4.17500000000e+0 | 5.42575197531e-2 | 5.42575197531e-2 |
| 4.20000000000e+0 | 4.20000000000e+0 | 5.36480686695e-2 | 5.36480686695e-2 |
| 4.22500000000e+0 | 4.22500000000e+0 | 5.30486389709e-2 | 5.30486389709e-2 |
| 4.25000000000e+0 | 4.25000000000e+0 | 5.24590163934e-2 | 5.24590163934e-2 |
| 4.27500000000e+0 | 4.27500000000e+0 | 5.18789922506e-2 | 5.18789922506e-2 |
| 4.30000000000e+0 | 4.30000000000e+0 | 5.13083632632e-2 | 5.13083632632e-2 |
| 4.32500000000e+0 | 4.32500000000e+0 | 5.07469313965e-2 | 5.07469313965e-2 |
| 4.35000000000e+0 | 4.35000000000e+0 | 5.01945037018e-2 | 5.01945037018e-2 |
| 4.37500000000e+0 | 4.37500000000e+0 | 4.96508921645e-2 | 4.96508921645e-2 |
| 4.40000000000e+0 | 4.40000000000e+0 | 4.91159135560e-2 | 4.91159135560e-2 |
| 4.42500000000e+0 | 4.42500000000e+0 | 4.85893892921e-2 | 4.85893892921e-2 |
| 4.45000000000e+0 | 4.45000000000e+0 | 4.80711452950e-2 | 4.80711452950e-2 |
| 4.47500000000e+0 | 4.47500000000e+0 | 4.75610118605e-2 | 4.75610118605e-2 |
| 4.50000000000e+0 | 4.50000000000e+0 | 4.70588235294e-2 | 4.70588235294e-2 |
| 4.52500000000e+0 | 4.52500000000e+0 | 4.65644189634e-2 | 4.65644189634e-2 |
| 4.55000000000e+0 | 4.55000000000e+0 | 4.60776408248e-2 | 4.60776408248e-2 |
| 4.57500000000e+0 | 4.57500000000e+0 | 4.55983356607e-2 | 4.55983356607e-2 |
| 4.60000000000e+0 | 4.60000000000e+0 | 4.51263537906e-2 | 4.51263537906e-2 |
| 4.62500000000e+0 | 4.62500000000e+0 | 4.46615491975e-2 | 4.46615491975e-2 |
| 4.65000000000e+0 | 4.65000000000e+0 | 4.42037794231e-2 | 4.42037794231e-2 |
| 4.67500000000e+0 | 4.67500000000e+0 | 4.37529054664e-2 | 4.37529054664e-2 |
| 4.70000000000e+0 | 4.70000000000e+0 | 4.33087916847e-2 | 4.33087916847e-2 |
| 4.72500000000e+0 | 4.72500000000e+0 | 4.28713056992e-2 | 4.28713056992e-2 |
| 4.75000000000e+0 | 4.75000000000e+0 | 4.24403183024e-2 | 4.24403183024e-2 |
| 4.77500000000e+0 | 4.77500000000e+0 | 4.20157033691e-2 | 4.20157033691e-2 |
| 4.80000000000e+0 | 4.80000000000e+0 | 4.15973377704e-2 | 4.15973377704e-2 |
| 4.82500000000e+0 | 4.82500000000e+0 | 4.11851012896e-2 | 4.11851012896e-2 |
| 4.85000000000e+0 | 4.85000000000e+0 | 4.07788765420e-2 | 4.07788765420e-2 |
| 4.87500000000e+0 | 4.87500000000e+0 | 4.03785488959e-2 | 4.03785488959e-2 |
| 4.90000000000e+0 | 4.90000000000e+0 | 3.99840063974e-2 | 3.99840063974e-2 |
| 4.92500000000e+0 | 4.92500000000e+0 | 3.95951396966e-2 | 3.95951396966e-2 |
| 4.95000000000e+0 | 4.95000000000e+0 | 3.92118419763e-2 | 3.92118419763e-2 |
| 4.97500000000e+0 | 4.97500000000e+0 | 3.88340088833e-2 | 3.88340088833e-2 |
| 5.00000000000e+0 | 5.00000000000e+0 | 3.84615384615e-2 | 3.84615384615e-2 |
On a narrow screen, focus or swipe within a figure or table to scroll horizontally. Labels retain their designed size.
| r | S(0) | ξ/a | ka at half height |
|---|---|---|---|
| 1/25 | 25 | 5 | 1/5 |
| 4/25 | 25/4 | 5/2 | 2/5 |
| 1 | 1 | 1 | 1 |
Further: derive the scaling free energy
For an ordinary finite-temperature continuous transition, define the signed reduced temperature t=(T−T_c)/T_c. Let h be a dimensionless field conjugate to the order parameter, not necessarily the laboratory field in tesla. In fixed microscopic length and energy units, write the singular free-energy density as f_s(t,h,{g_i})=b^(−d)f_s(t b^(y_t),h b^(y_h),{g_i b^(y_i)}). Analytic background terms are separate.
Assume a single thermal relevant field, conventional isotropic short-range scaling and no dangerously irrelevant variable spoiling this reduction. At h=0 choose b=|t|^(−1/y_t). The flow stops when the rescaled temperature-like field is of order 1, giving ξ proportional to |t|^(−1/y_t), so ν=1/y_t. The same argument gives f_s proportional to |t|^(dν), with possibly different amplitudes on the two sides.
Differentiate with respect to the conjugate field: m=−∂f_s/∂h scales as b^(−d+y_h), and χ=∂m/∂h scales as b^(−d+2y_h). The spontaneous ordered branch gives β_m=(d−y_h)/y_t; the zero-field susceptibility gives γ=(2y_h−d)/y_t. The subscript in β_m distinguishes this exponent from inverse temperature β=1/(k_B T).
At t=0 choose b=|h|^(−1/y_h) to obtain m proportional to sign(h)|h|^(1/δ), with δ=y_h/(d−y_h), when that denominator is positive. The field dimension Δ_φ=d−y_h=(d−2+η)/2 gives η=d+2−2y_h. Two thermal derivatives yield α=2−dν, the hyperscaling relation under the stated assumptions. Marginal logarithms, special dimensions and a vanishing power can require more than a simple power law.
| Exponent | Behaviour or meaning | Scaling relation |
|---|---|---|
| ν | ξ ∝ |t|^(−ν) | 1/y_t |
| β_m | Spontaneous |m| ∝ (−t)^(β_m) | (d−y_h)/y_t |
| γ | χ ∝ |t|^(−γ) | (2y_h−d)/y_t |
| δ | At t=0, |m| ∝ |h|^(1/δ) | y_h/(d−y_h) |
| η | Critical G(R) ∝ R^(−d+2−η) | d+2−2y_h |
| α | Singular heat capacity power | 2−dν, if hyperscaling applies |
Further: count Gaussian engineering dimensions
Use the dimensionless scalar action S[φ]=∫d^d x [(∇φ)²/2+rφ²/2+uφ⁴/4!−hφ], with coordinates in declared microscopic length units and a cutoff. At the massless Gaussian fixed point r=u=h=0, preserve the gradient term under x′=x/b by defining φ′(x′)=b^((d−2)/2)φ(x). The Gaussian field scaling dimension is Δ_φ=(d−2)/2.
Substitution into the other terms gives r′=b²r, u′=b^(4−d)u and h′=b^((d+2)/2)h. Thus the quartic coupling is relevant for d<4, marginal at linear order for d=4, and irrelevant for d>4. These are local engineering dimensions at the Gaussian fixed point, not exact eigenvalues at every interacting fixed point.
Our assigned initial r=0.01 and u=0.001 illustrate those powers. At b=4, r has grown to 0.16 in every displayed dimension, while the quartic values differ as shown. At b=8, r=0.64: continuing the formal power is algebraically possible, but interpreting a perturbation as small is no longer well justified. The table does not pretend to be a solved nonlinear RG trajectory.
Count local powers without inventing a nonlinear flow
The four assigned scale factors illustrate the Gaussian engineering rules. They do not integrate an interacting beta function or supply a material calibration.
Mass perturbation grows in every displayed dimension
- r(b), all displayed dimensions
Quartic relevance depends on dimension
- d=2, yᵤ=2
- d=3, yᵤ=1
- d=4, yᵤ=0; linear-marginal only
- d=5, yᵤ=-1
- At d=4 the displayed quartic rule is marginal only at linear order. Nonlinear terms and logarithms are not computed by this constant row.
- Lines join the four assigned scale samples. They are not trajectories in time or a solved Wilson–Fisher phase portrait.
- The conventional exponent and hyperscaling table remains conditional. A dangerously irrelevant stabilizing coupling cannot be discarded from every observable.
The original whole-table links and complete additional values are below.
All assigned Gaussian engineering-flow controls, not a solved nonlinear trajectory. r and u are rounded to 12 significant digits; eigenvalues and field dimensions are exact.
| d | Spatial scale b | yᵣ | yᵤ | yₕ | Δφ | r(b)≈ | u(b)≈ |
|---|---|---|---|---|---|---|---|
| 2 | 1 | 2 | 2 | 2 | 0 | 1.00000000000e-2 | 1.00000000000e-3 |
| 2 | 2 | 2 | 2 | 2 | 0 | 4.00000000000e-2 | 4.00000000000e-3 |
| 2 | 4 | 2 | 2 | 2 | 0 | 1.60000000000e-1 | 1.60000000000e-2 |
| 2 | 8 | 2 | 2 | 2 | 0 | 6.40000000000e-1 | 6.40000000000e-2 |
| 3 | 1 | 2 | 1 | 5/2 | 1/2 | 1.00000000000e-2 | 1.00000000000e-3 |
| 3 | 2 | 2 | 1 | 5/2 | 1/2 | 4.00000000000e-2 | 2.00000000000e-3 |
| 3 | 4 | 2 | 1 | 5/2 | 1/2 | 1.60000000000e-1 | 4.00000000000e-3 |
| 3 | 8 | 2 | 1 | 5/2 | 1/2 | 6.40000000000e-1 | 8.00000000000e-3 |
| 4 | 1 | 2 | 0 | 3 | 1 | 1.00000000000e-2 | 1.00000000000e-3 |
| 4 | 2 | 2 | 0 | 3 | 1 | 4.00000000000e-2 | 1.00000000000e-3 |
| 4 | 4 | 2 | 0 | 3 | 1 | 1.60000000000e-1 | 1.00000000000e-3 |
| 4 | 8 | 2 | 0 | 3 | 1 | 6.40000000000e-1 | 1.00000000000e-3 |
| 5 | 1 | 2 | -1 | 7/2 | 3/2 | 1.00000000000e-2 | 1.00000000000e-3 |
| 5 | 2 | 2 | -1 | 7/2 | 3/2 | 4.00000000000e-2 | 5.00000000000e-4 |
| 5 | 4 | 2 | -1 | 7/2 | 3/2 | 1.60000000000e-1 | 2.50000000000e-4 |
| 5 | 8 | 2 | -1 | 7/2 | 3/2 | 6.40000000000e-1 | 1.25000000000e-4 |
On a narrow screen, focus or swipe within a figure or table to scroll horizontally. Labels retain their designed size.
| Spatial dimension d | Quartic y_u=4−d | r(b=4) | u(b=4) |
|---|---|---|---|
| 2 | 2 | 0.16 | 0.016 |
| 3 | 1 | 0.16 | 0.004 |
| 4 | 0 | 0.16 | 0.001 |
| 5 | −1 | 0.16 | 0.00025 |
Further: keep a dangerously irrelevant interaction
The Gaussian upper critical dimension for the short-range scalar φ⁴ problem is 4. Below it, the quartic interaction grows and generic critical behaviour is not captured by simply discarding it. At d=4, nonlinear flow produces logarithmic corrections in the usual scalar theory; a zero linear eigenvalue is not proof of an exactly constant coupling at all scales.
Above 4, the quartic interaction can shrink under the long-distance flow and still be essential in an observable. In a uniform mean-field ordered state with r<0 and u>0, minimizing rφ²/2+uφ⁴/24 gives m²=−6r/u. Setting u=0 before this minimization removes the stabilizing term and destroys the ordered solution. The denominator makes this a dangerously irrelevant variable for that question.
Consequently, inserting Gaussian y_t and y_h into every hyperscaling formula above 4 gives incorrect order-parameter predictions. The usual mean-field β_m=1/2 and δ=3 require retaining the quartic interaction in the ordered equation of state. Regular background heat capacity, Gaussian fluctuation contributions and the singular thermodynamic part must also be distinguished. One cannot extend α=2−dν or ordinary finite-size formulas beyond their conditions by a formal substitution alone.
The ε expansion offers a controlled return path near d=4−ε, with small positive ε. The interacting Wilson–Fisher fixed point has a quartic coupling of order ε, and for the one-component short-range theory y_t=2−ε/3+O(ε²), hence ν=1/2+ε/12+O(ε²). The anomalous exponent starts at order ε². Setting ε=1 in a first-order expression is an extrapolation, not an exact three-dimensional exponent or a small-error bound.
Further: test finite-size scaling and corrections
A finite system can show a sharp-looking rounded peak without a thermodynamic singularity. For a conventional continuous transition, with fixed shape and boundary conditions in the scaling regime, a useful form is f_s(t,h,L)=L^(−d)𝓕(tL^(1/ν),hL^(y_h),{g_iL^(y_i)}), with L measured in microscopic length units. Irrelevant fields create corrections that vanish only asymptotically; analytic backgrounds may remain important.
For example, the singular susceptibility then has χ_s(t,L)=L^(γ/ν)𝓧(tL^(1/ν)) at zero field, up to corrections. A collapse plots χ_s/L^(γ/ν) against tL^(1/ν) for several sizes. Useful evidence includes uncertainty estimates, increasing-size stability, controlled backgrounds and boundaries, and corrections consistent with the fit. A visually pleasing collapse obtained by adjusting enough parameters is not independent proof of the exponents.
Finite-size rounding and the position of a peak need not be identical effects. A peak can be displaced from the infinite-system T_c. Above the upper critical dimension, dangerously irrelevant variables modify the simple finite-size prescription and boundary conditions matter. No finite-size numerical dataset or fitted exponent is supplied in this lesson; the exact Gaussian collapse above tests its assigned equation, not this general fitting method.
Use static RG without inventing a clock
Critical correlations connect fluctuation measurements, scattering at small wave number and comparisons between systems that share a long-distance description. An RG calculation also explains why a continuum theory can work without resolving every microscopic interaction, while specifying which perturbations cannot be neglected.
Statics alone does not supply a relaxation time. A relation such as τ proportional to ξ^z needs an additional dynamics and its dynamic exponent z; conserved and nonconserved order parameters need not have the same dynamics. The scale factor b is not elapsed time, and a sequence of coupling values is not a movie of a cooling sample.
Return to the chain: its physical equilibrium length survived exact elimination of intermediate spins. The lesson is not that every coarse model preserves every microscopic fact. It is that a careful scale transformation names what is retained, what is summed over, how normalization changes, and which units make successive descriptions comparable.
Worked examples
A shorter number, not a shorter length
For q=1/2, what happens to ξ after one exact decimation?
- Use ξ/a=−1/ln(1/2)=1/ln2, about 1.4426950.
- The retained-neighbour correlation is q′=1/4 and the new spacing is a′=2a.
- Thus ξ/a′=−1/ln(1/4)=1/(2ln2), about 0.72134752. Restore a′ before comparing physical lengths.
ξ=(2a)/(2ln2)=a/ln2. Only the numerical length measured in retained sites has halved.
Recover the removed spin’s normalization
At q=1/2, find K′ and A, and check opposed endpoints.
- K=atanh(1/2)=(ln3)/2, and cosh2K=(3+1/3)/2=5/3.
- Therefore K′=(1/2)ln(5/3), and A=2√(5/3).
- For opposed endpoints, A exp(−K′)=2√(5/3)/√(5/3)=2, the sum of two unit weights.
K′=(1/2)ln(5/3) and A=2√(5/3). Omitting A would fail this local identity.
An eight-spin partition check
What factor multiplies the retained four-spin partition function at q=1/2?
- Four middle spins are summed out, so Z₈=A⁴Z₄(K′).
- Here A=2√(5/3), so A⁴=16(5/3)²=400/9.
- The original sum has 256 configurations, and the retained sum 16. The factor accounts for their weights, not a replacement count of equally likely states.
Z₈(K)=(400/9)Z₄(K′). The full positive weighted sums agree exactly.
Read a static peak width
For r=1/25, find S(0), ξ/a and ka at half height.
- S(0)=1/r=25, and ξ/a=1/√r=5.
- Half height requires 1/[r+(ka)²]=(1/2)(1/r).
- Thus (ka)²=r and ka=1/5 for nonnegative k; equivalently kξ=1.
S(0)=25, ξ/a=5 and half height occurs at ka=0.2. These are dimensionless model quantities, not measured units.
The same scale change in two dimensions of theory
Compare u at b=4 for d=3 and d=5, beginning at u₀=0.001.
- The local Gaussian rule is u(b)=b^(4−d)u₀.
- For d=3, multiply by 4: u=0.004.
- For d=5, divide by 4: u=0.00025. These classify a perturbation around the Gaussian point, not a complete interacting flow.
The perturbation grows in d=3 and shrinks in d=5 under the declared longer-length convention.
Derive the susceptibility exponent
Why does χ scale with 2y_h−d rather than y_h−d?
- The singular free-energy density has the overall factor b^(−d).
- One derivative with respect to h introduces b^(y_h); a second introduces another b^(y_h).
- Set b=|t|^(−1/y_t), yielding χ_s proportional to |t|^(−(2y_h−d)/y_t).
γ=(2y_h−d)/y_t under conventional scaling. Susceptibility is the second field derivative of the free energy, not the order parameter itself.
Why a shrinking coupling can remain necessary
Minimize the uniform scalar potential r m²/2+u m⁴/24 for r<0 and u>0.
- Its derivative is m(r+u m²/6). The nonzero stationary solutions obey m²=−6r/u.
- The curvature at those solutions is r+u m²/2=−2r>0, so they are local minima.
- The positive quartic term stabilizes large |m|. Taking u to zero before minimizing leaves a negative quadratic unbounded below.
The ordered minima depend on 1/√u. Irrelevance of u near the Gaussian point is not permission to discard it in this observable.
Do not invent a finite-T exponent
Does δ′≈2δ near q=1 imply ξ∝T^(−1) for the chain?
- δ=1−tanhK is a particular dimensionless coupling distance, not T itself.
- At large K, δ≈2exp(−2K), and −lnq≈δ.
- Therefore ξ/a≈1/δ≈exp[2J/(k_B T)]/2, not 1/T.
No. The zero-temperature divergence is exponential in 1/T; dividing by T_c=0 to define an ordinary reduced temperature would also be invalid.
Try it yourself
In the opening chain, what is C(4)?
Show the answer and reasoning
1/16.
The physical separation is 4a and C(r)=(1/2)^r. This is an ensemble correlation, not a probability density in r.
Which original separation corresponds to one retained bond after two decimations?
Show the answer and reasoning
4a.
Each operation doubles the spacing; two operations give a₂=4a.
Why subtract ⟨s_i⟩⟨s_j⟩ in a connected correlation?
Show the answer and reasoning
To remove the contribution due solely to the mean values.
A nonzero ordered mean can keep ⟨s_i s_j⟩ nonzero at long distance without representing a long connected fluctuation.
What two weights are summed for aligned endpoints in the local chain piece?
Show the answer and reasoning
exp(2K) and exp(−2K).
The middle spin is aligned with both endpoints or opposed to both, giving the sum 2cosh2K.
May A be dropped when calculating free energy from Z?
Show the answer and reasoning
Not without restoring its contribution.
For even N, lnZ_N=(N/2)lnA+lnZ_(N/2). Since A depends on K, thermodynamic derivatives can be affected.
For q=4/5, what is q after one decimation?
Show the answer and reasoning
16/25.
The exact rule q′=q² lowers every q strictly between 0 and 1 under the longer-length flow.
Does q=1 represent a positive critical temperature for this model?
Show the answer and reasoning
No; it is the T→0 boundary.
K=J/(k_BT) tends to infinity as T approaches 0, and the exact chain has no intermediate finite-T fixed point.
Is b=8 eight seconds of relaxation?
Show the answer and reasoning
No.
b is a dimensionless spatial scale factor. An equilibrium scale transformation does not specify an evolution law or a clock.
For r=4/25, what are ξ/a and S(0)?
Show the answer and reasoning
ξ/a=5/2 and S(0)=25/4.
Apply 1/√r and 1/r, keeping the assigned dimensionless amplitude fixed.
Must equal ka grids have equal kξ endpoints for all supplied r?
Show the answer and reasoning
No.
kξ=(ka)/√r. Different ξ values map the same ka endpoint to different scaled horizontal ranges.
If y_u=0 at d=4, is u exactly marginal?
Show the answer and reasoning
The linear calculation alone cannot decide.
Higher-order beta-function terms are needed. In the usual scalar φ⁴ critical theory they lead to logarithmic corrections.
At b=8, what is r from r₀=0.01, and what qualification is needed?
Show the answer and reasoning
r=0.64 by the formal engineering rule.
Multiplying by b² is algebraically defined, but a perturbation this large is no longer safely described as very close to the Gaussian point.
Can α=2−dν be used automatically above the upper critical dimension?
Show the answer and reasoning
No.
The conventional derivation neglects variables that may be dangerously irrelevant. The ordered scalar theory retains u in a denominator.
For a scale change b=2 with a thermal eigenvalue λ_t=4, what is y_t and the conventional ν?
Show the answer and reasoning
y_t=2 and ν=1/2.
λ_t=b^(y_t), so y_t=ln4/ln2. The ν inference additionally requires the stated ordinary thermal scaling conditions; it is not the opening chain’s finite-T law.
What powers of b multiply m and χ if f_s has the conventional homogeneity form?
Show the answer and reasoning
m scales as b^(−d+y_h); χ scales as b^(−d+2y_h).
Each derivative with respect to the conjugate field contributes one power b^(y_h).
A small set of rounded finite-size curves collapse after several fitted adjustments. Is that alone proof of critical exponents?
Show the answer and reasoning
No.
Test larger sizes, uncertainties, backgrounds, boundary conditions and correction terms. Fitting a visual collapse is not an independent validation of its assumptions.
What to carry forward
- The exact chain preserves its physical correlation length when the new lattice spacing is restored; the numerical count in rescaled sites changes.
- Summing short-distance variables changes couplings and normalization. A tractable truncation is not automatically an exact transformation.
- Relevance is a local statement near a fixed point with a declared longer-length convention; marginal directions need nonlinear analysis.
- Universality compares appropriate long-distance behaviour, not every amplitude, microscopic parameter or visually similar curve.
- Critical-exponent and hyperscaling formulas have conditions. Zero-temperature chains, marginal logarithms and dangerously irrelevant variables require their own limits.
- Finite-size and static Gaussian collapse can be informative without proving an interacting universality class or a dynamic relaxation law.
References
Sources for the concepts and teaching approaches in this article.
- Statistical Mechanics II: Statistical Physics of Fields (MIT OpenCourseWare, Spring 2014)Mehran KardarCourse notesStatistical Mechanics II: Statistical Physics of Fields (MIT OpenCourseWare, Spring 2014)Lecture 7: the renormalization groupLecture 13: exact one-dimensional decimation
- Statistical Field Theory: The Renormalisation GroupDavid TongCourse notes
- Statistical Mechanics I lecture notes (2025)Naoki KawashimaCourse notesStatistical Mechanics I lecture notes (2025)Lecture 4: Ornstein–Zernike formulaLecture 8: more consequences of the renormalization group