Hello Wolfram Community,
I want to share a computational observation that I have not seen discussed in
this precise form, and I would genuinely welcome input on whether there is an
obvious derivation I am missing.
The observation
Consider the coefficient:
Gcurvature[d_] := Gamma[1 + d]^2 / Gamma[1 + 2 d]
For integer arguments, this equals 1/Binomial[2n, n] — the reciprocal of the
central binomial coefficients. The first few values:
Gcurvature(1) = 1/2, Gcurvature(2) = 1/6, Gcurvature(3) = 1/20, Gcurvature(4) = 1/70
I have observed that the critical exponents of the three-dimensional Ising
universality class appear to be expressible as exact rationals built from these
four coefficients.
The three main expressions
eta3D = Gcurvature[3]^3 / (Gcurvature[1]^3 * Gcurvature[2]^2) = 9/250 = 0.0360
nu3D = Gcurvature[3]^3 / (Gcurvature[1] * Gcurvature[2]^2 * Gcurvature[4]) = 63/100 = 0.6300
beta3D = Gcurvature[4]^2 / (Gcurvature[1]^2 * Gcurvature[3]^2) = 16/49 = 0.326531...
The numerator of nu3D simplifies remarkably: 5040/8000 = 7!/20^3.
Comparison with the conformal bootstrap
The state-of-the-art values from Chester et al. (2024, arXiv:2411.15300) are:
eta = 0.0362976(5) — GOD prediction 9/250 — error 0.82%
nu = 0.6299710(4) — GOD prediction 63/100 — error 0.005%
beta = 0.3264187(6) — GOD prediction 16/49 — error 0.034%
By standard scaling relations, the remaining exponents also become exact rationals:
gamma = 30933/25000 (bootstrap: 1.237076, error 0.020%)
alpha = 11/100 (bootstrap: 0.110087, error 0.079%)
delta = 1241/259 (bootstrap: 4.789843, error 0.035%)
The 2D case is exact
For the 2D Ising model (Onsager 1944), the same framework gives exact results:
eta_2D = Gcurvature[1]^2 = 1/4 (Onsager: 1/4) EXACT
nu_2D = Gcurvature[0] = 1 (Onsager: 1) EXACT
beta_2D = Gcurvature[1]^3 = 1/8 (Onsager: 1/8) EXACT
Mean-field values (d >= 4) also match exactly:
eta_MF = 0 EXACT
nu_MF = Gcurvature[1] = 1/2 EXACT
beta_MF = Gcurvature[1] = 1/2 EXACT
Context
This observation forms part of a broader framework I have been developing
(God Programme manifiesto), in which these coefficients
arise as curvature coefficients of a fractal algebra built on fractional
derivatives (GOD Programme VIII: Critical Exponents of the Ising Model from the Torrado Algebra). However, the numerical match above holds independently of that
framework and can be verified by anyone with the attached notebook.
What I am asking
The 2D Ising match and the mean-field match are exact. The 3D match is
empirical at the 0.005%-0.82% level. I do not have a rigorous derivation
of the exponent patterns (why specifically Gcurvature[3]^3 / (Gcurvature[1]^3 * Gcurvature[2]^2)
for eta, for example).
My question to the Community: does anyone see an obvious path to this via
the renormalisation group, conformal field theory, or the structure of
central binomial coefficients? Has any similar pattern been observed?
The attached notebook contains all computations in roughly 30 lines of
Wolfram Language. Please feel free to run it, criticize it, or extend it, even checking if the algebra behind the coefficients is properly built.
Thanks in advance.
Francisco Torrado Cano
Independent Researcher
Cáceres, Spain
GOD_Art8_CriticalExponents.nb (1.7 KB)