Annealed central limit theorems for the Ising model on random graphs

TL;DR

This paper proves annealed central limit theorems for the magnetization in various Ising models on random graphs, identifying phase transition regimes and leveraging reductions to simpler models for analysis.

Contribution

It establishes annealed CLTs for Ising models on complex random graphs, including the generalized random graph and configuration models, with explicit phase transition analysis.

Findings

CLT holds in the uniqueness regime for generalized random graphs.

CLT holds for all parameters in degree-1 and degree-2 configuration models.

Existence of a finite annealed critical inverse temperature for generalized random graphs.

Abstract

The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by $\sqrt{N}$ of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature $0 \leq \beta^{\mathrm \scriptscriptstyle an}_c < \infty$ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature $\beta$ and external magnetic field $B$ for which either $\beta < \beta^{\mathrm \scriptscriptstyle an}_c$ and $B=0$, or $\beta>0$ and $B \neq 0$. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters $\beta$ and $B$, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.