Fluctuations of the Ising free energy on Erd\H{o}s-R\'enyi graphs

TL;DR

This paper characterizes the limiting distribution of the log-partition function of the ferromagnetic Ising model on Erdős-Rényi graphs, revealing Gaussian fluctuations under certain conditions and bounded variance in others.

Contribution

It provides a detailed analysis of the fluctuations of the Ising free energy on Erdős-Rényi graphs, including the derivation of limiting distributions in different regimes.

Findings

Gaussian fluctuations when B>0 or B=0 with d>1 and β>ath(1/d)

Bounded variance and infinite sum structure when B=0 and d≤1 or β<ath(1/d)

Limiting distributions described by stochastic fixed point problems

Abstract

We investigate the ferromagnetic Ising model on the Erd\H{o}s-R\'enyi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(\beta,B)$, where $Z_{\mathbb{G}(n,m)}(\beta,B)$ is the partition function at inverse temperature $\beta>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $\beta>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $\Theta(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $\beta<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.