The Ising Model on a Two-Community Stochastic Block Model

TL;DR

This paper characterizes the phase transitions of the Ising model on a two-community stochastic block model, analyzing magnetization behavior and proving a quenched CLT in different regimes.

Contribution

It provides a complete phase diagram and detailed analysis of magnetization laws, including convergence to mixtures and fluctuation results, for the Ising model on this graph.

Findings

Identifies the phase transition between uniqueness and non-uniqueness of the Gibbs measure.

Shows the magnetization law converges to mixtures supported on two or four points depending on parameters.

Proves a quenched central limit theorem for magnetization fluctuations in the subcritical regime.

Abstract

We study the Ising model on a two-community stochastic block model, where $n$ spins are split into two equal groups with inter-community interaction parameter $\alpha_n\in[0,1]$. We provide a complete characterization of the phase diagram and show that, almost surely with respect to the graph realization, the model undergoes a uniqueness/non-uniqueness phase transition of the Gibbs measure. In particular, in the supercritical regime, the law of the magnetization vector of the two communities converges to a mixture of Dirac measures that, depending on whether $\alpha_n\gg 1/n$ or $\alpha_n\lesssim1/n$, is supported on two or four points, with possibly different weights. In the uniqueness region, we further analyze the fluctuations of the magnetization vector in the subcritical regime and we prove a quenched central limit theorem.