Fisher Curvature Scaling at Critical Points: An Exact Information-Geometric Exponent from Periodic Boundary Conditions

Maxim Zhuravlev

Published Mar 10, 2026• 0 citations

Abstract

Proposes a new critical exponent d_R for Fisher curvature scaling and proves d_R = (dnu + 2eta)/(d*nu + eta). Validated on 2D Ising (exact TM L=3-9, MCMC L=10-24), 3D Ising, 2D Potts q=3,4. Published on arXiv as 2603.07651.

DOI: 10.5281/zenodo.18942480

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Fisher Curvature Scaling at Critical Points: An Exact Information-Geometric Exponent from Periodic Boundary Conditions

Maxim Zhuravlev

Published Mar 10, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.18942480

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Feb 27, 20260 citations

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arXiv:cond-mat/0103332Physical review. B, Condensed matterJan 24, 200266 citations

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.