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 dN regular graph GN on N vertices. In particular, if GN is "well connected", we show that whenever dN ≫ √ N , the fluctuations are universal and same as that of the Curie-Weiss model in the entire Ferro-magnetic parameter regime. We give a counterexample to demonstrate that the condition dN ≫ √ N is tight, in the sense that the limiting distribution changes if dN ∼ √ N except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to dN ≫ N 1/3 . Our results conclude 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 non-negative 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, and tight error bounds for the Mean-Field approximation of the partition function.