Fluctuations of the Magnetization for Ising models on Erd\H{o}s-R\'enyi Random Graphs -- the Regimes of Low Temperature and External Magnetic Field

TL;DR

This paper analyzes the fluctuations of magnetization in Ising models on Erdős-Rényi graphs, focusing on low temperature and external magnetic fields, proving a CLT and describing log-partition function fluctuations under certain conditions.

Contribution

It extends previous work by establishing a quenched CLT for magnetization and characterizing log-partition function fluctuations in low temperature and external field regimes.

Findings

Proves a quenched Central Limit Theorem for magnetization.

Describes fluctuations of the log-partition function.

Analyzes regimes with $p^3N o abla$.

Abstract

We continue our analysis of Ising models on the (directed) Erd\H{o}s-R\'enyi random graph $G(N,p)$. We prove a quenched Central Limit Theorem for the magnetization and describe the fluctuations of the log-partition function. In the current note we consider the low temperature regime $\beta>1$ and the case when an external magnetic field is present. In both cases, we assume that $p=p(N)$ satisfies $p^3N \to \infty$.