Quenched central limit theorems for the Ising model on random graphs

TL;DR

This paper establishes quenched and averaged quenched central limit theorems for the magnetization of the Ising model on various random graphs in the uniqueness regime, extending understanding of fluctuations in these systems.

Contribution

It proves new quenched and averaged quenched CLTs for the Ising model on random graphs, including general tree-like graphs and specific models with explicit computations.

Findings

Quenched CLTs hold for general tree-like graphs.

Averaged quenched CLTs are established for 2-regular and degree-1/2 configuration models.

Results demonstrate Gaussian fluctuations of magnetization in the specified regimes.

Abstract

The main goal of the paper is to prove central limit theorems for the magnetization rescaled by $\sqrt{N}$ for the Ising model on random graphs with $N$ vertices. Both random quenched and averaged quenched measures are considered. We work in the uniqueness regime $\beta>\beta_c$ or $\beta>0$ and $B\neq0$, where $\beta$ is the inverse temperature, $\beta_c$ is the critical inverse temperature and $B$ is the external magnetic field. In the random quenched setting our results apply to general tree-like random graphs (as introduced by Dembo, Montanari and further studied by Dommers and the first and third author) and our proof follows that of Ellis in $\mathbb{Z}^d$. For the averaged quenched setting, we specialize to two particular random graph models, namely the 2-regular configuration model and the configuration model with degrees 1 and 2. In these cases our proofs are based on explicit computations relying on the solution of the one dimensional Ising models.