The Information Geometry of the Ising Model on Planar Random Graphs

TL;DR

This paper explores the information geometric properties of the Ising model on planar random graphs, revealing how the scalar curvature diverges near the phase transition and analyzing the impact of critical exponents.

Contribution

It provides an exact calculation of the scalar curvature's scaling behavior for the Ising model on planar random graphs, extending the understanding of information geometry in complex systems.

Findings

Scalar curvature diverges as |β - β_c|^{-2} near criticality

Discrepancy explained by negative critical exponent α

Supports the scaling relation of scalar curvature at phase transitions

Abstract

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterisation of the phase structure, particularly in the case where there are two such parameters -- such as the Ising model with inverse temperature $\beta$ and external field $h$. In various two parameter calculable models the scalar curvature ${\cal R}$ of the information metric has been found to diverge at the phase transition point $\beta_c$ and a plausible scaling relation postulated: ${\cal R} \sim |\beta- \beta_c|^{\alpha - 2}$. For spin models the necessity of calculating in non-zero field has limited analytic consideration to 1D, mean-field and Bethe lattice Ising models. In this letter we use the solution in field of the Ising model on an ensemble of planar random graphs (where $\alpha=-1, \beta=1/2, \gamma=2$) to evaluate the scaling behaviour of the scalar curvature, and find ${\cal R} \sim | \beta- \beta_c |^{-2}$. The apparent discrepancy is traced back to the effect of a negative $\alpha$.