Fluctuations of the Magnetization for Ising Models on Dense Erd\H{o}s-R\'enyi Random Graphs

TL;DR

This paper proves a quenched Central Limit Theorem for the magnetization of Ising models on dense Erdős-Rényi graphs in the high-temperature regime, under specific conditions on the edge probability.

Contribution

It establishes a quenched CLT for magnetization in Ising models on dense Erdős-Rényi graphs, extending previous results to a new regime of edge probabilities.

Findings

Quenched CLT holds for high-temperature regime with p^3 N^2 → +∞

Results apply to directed Erdős-Rényi graphs

Extends understanding of fluctuations in Ising models on random graphs

Abstract

We analyze Ising/Curie-Weiss models on the (directed) Erd\H{o}s-R\'enyi random graph on $N$ vertices in which every edge is present with probability $p$. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime $\beta<1$ when $p=p(N)$ satisfies $p^3N^2\to +\infty$.