Fluctuations of the Ising free energy on Erdős-Rényi graphs

TL;DR

This work establishes a precise limiting distribution for the fluctuations of the Ising free energy on Erdős-Rényi graphs across key regimes. Using a refined Belief Propagation framework, correlated graph couplings, and a martingale CLT, it derives Gaussian fluctuations with variance given by fixed-point functionals in the presence of a field, while also detailing non-Gaussian, bounded fluctuations in the high-temperature zero-field case. The zero-field regime is further nuanced by a pure-state decomposition that recovers replica symmetry within each pure state and yields a Gaussian limit in the low-temperature phase, except at a countable set of discontinuities. In the high-temperature zero-field regime, small-subgraph conditioning provides a log-normal-type limit. Overall, the paper combines cavity-method heuristics, local weak convergence, and probabilistic conditioning techniques to deliver a thorough, rigorous picture of free-energy fluctuations for Ising models on sparse random graphs.

Abstract

We investigate the ferromagnetic Ising model on the Erdős-Rényi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(β,B)$, where $Z_{\mathbb{G}(n,m)}(β,B)$ is the partition function at inverse temperature $β>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $β>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $Θ(n)$ 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 $β<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.

Figures (3)

• Figure 1: Plots of $\beta\mapsto\Sigma(d,\beta)^2$ at $d =1.9$ and at external fields $B=0$, $B=0.25$, $B=0.5$ and $B=0.75$. The dashed vertical line marks the critical inverse temperature $\beta_{\mathrm f}(1.9)$.
• Figure 2: Plot of $\beta\mapsto {r_{d,\beta}}$ for $d=1$, $d=3$ and $d=5$.
• Figure 3: The distributions $\pi_{d,\beta,0,t}^\otimes$ for $d=1.9$, $\beta=0.65$, and $t=0.1, 0.5, 0.9$. The atom at $(0,0)$ has been removed to improve visibility.

Theorems & Definitions (147)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1: Dembo_2010
• Lemma 2.2: e.g. Brasil
• Lemma 2.3
• Corollary 2.4
• Proposition 2.5
• Remark 2.6