Fluctuations in mean-field Ising models

Abstract

In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately ${\mathit{d}}_{\mathit{N}}$ regular graph ${\mathit{G}}_{\mathit{N}}$ on N vertices. In particular, if ${\mathit{G}}_{\mathit{N}}$ satisfies a “spectral gap” condition, we show that whenever ${\mathit{d}}_{\mathit{N}}\gg \sqrt{\mathit{N}}$, the fluctuations are universal and the same as that of the Curie–Weiss model in the entire ferromagnetic parameter regime. We give a counterexample to demonstrate that the condition ${\mathit{d}}_{\mathit{N}}\gg \sqrt{\mathit{N}}$ is tight, in the sense that the limiting distribution changes if ${\mathit{d}}_{\mathit{N}}\sim \sqrt{\mathit{N}}$ except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to ${\mathit{d}}_{\mathit{N}}\gg {\mathit{N}}^{1/3}$. Our results include universal fluctuations of the average magnetization in Ising models on regular graphs, Erdős–Rényi graphs (directed and undirected), stochastic block models, and sparse regular graphons. In fact, our results apply to general matrices with nonnegative entries, including Ising models on a Wigner matrix, and the block spin Ising model. As a by-product of our proof technique, we obtain Berry–Esseen bounds for these fluctuations, exponential concentration for the average of spins, tight error bounds for the mean-field approximation of the partition function, and tail bounds for various statistics of interest.