Fisher Curvature Scaling at Critical Points: An Exact Information-Geometric Exponent

Maxim Zhuravlev

Published Feb 27, 2026• 0 citations

Abstract

Introduces a novel scaling exponent d_R = (dnu + 2eta)/(d*nu + eta) for scalar curvature on per-bond Fisher manifolds at criticality. Confirmed to 4 significant figures for 2D Ising (d_R = 10/9). Extended predictions for 3D Ising, Potts q=3, and Potts q=4 validated numerically.

DOI: 10.5281/zenodo.18807278

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Fisher Curvature Scaling at Critical Points: An Exact Information-Geometric Exponent

Maxim Zhuravlev

Published Feb 27, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.18807278

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We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a

d

-dimensional lattice with periodic boundary conditions and

n = L^d

sites, the Fisher manifold has

m = d \cdot n

dimensions (one per bond), and we find

|\mathcal{R}(J_c)| \sim n^{d_R}

with

d_R = (dν+ 2η)/(dν+ η)

, where

ν

and

η

are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (

ν= 1

η= 1/4

), this predicts

d_R = 10/9

, confirmed by exact transfer-matrix computations (

L = 6

9

d_R = 1.1115 \pm 0.0002

) and multi-seed MCMC through

L = 24

. For 3D Ising (

ν= 0.630

η= 0.0363

), the prediction

d_R = 1.019

is consistent with MCMC on

L^3

tori up to

L = 10

(power-law fit:

d_R = 1.040

). For 2D Potts

q = 3

(predicted

33/29 \approx 1.138

), FFT-MCMC through

L = 40

shows

d_\mathrm{eff}

oscillating non-monotonically around

\sim 1.20

, consistent with

O(1/(\ln L)^2)

logarithmic corrections. For

q = 4

(predicted

22/19

), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity

R_3 = -R_1/2

R_4 = -R_2/2

holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.