Fluctuations in the Ising spin model on a sparse random graph

TL;DR

This paper analyzes the fluctuations of magnetization and overlaps in the dilute mean field ferromagnet, revealing Gaussian behavior and divergence of variance at the critical line.

Contribution

It provides a detailed computation of fluctuations in the Ising model on sparse graphs, highlighting Gaussian limits and critical behavior.

Findings

Magnetization fluctuations are Gaussian with diverging variance at criticality.

Multi-overlaps also tend to Gaussian variables with finite covariance at the critical line.

Results enhance understanding of phase transitions in sparse Ising models.

Abstract

We compute the fluctuations of the magnetization and of the multi-overlaps for the dilute mean field ferromagnet, in the high temperature region. The rescaled magnetization tends to a centered Gaussian variable with variance diverging at the critical line. The rescaled multi-overlaps also tend to centered independent Gaussian variables, but their covariance remain finite at the critical line.