The Information Geometry of the Ising Model on Planar Random Graphs

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 $β$ 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 $β_c$ and a plausible scaling relation postulated: ${\cal R} \sim |β- β_c|^{α- 2}$. 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 $α=-1, β=1/2, γ=2$) to evaluate the scaling behaviour of the scalar curvature, and find ${\cal R} \sim | β- β_c |^{-2}$. The apparent discrepancy is traced back to the effect of a negative $α$.