Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model

TL;DR

This paper analyzes the magnetization behavior of the dilute Curie-Weiss model on Erdős-Rényi graphs in the high-temperature regime, providing precise probabilistic results and limit theorems.

Contribution

It establishes sharp cumulant bounds and limit theorems for the magnetization of the annealed dilute Curie-Weiss model under specific asymptotic conditions.

Findings

Proves a central limit theorem with rate for magnetization.

Establishes a moderate deviation principle.

Provides concentration inequalities and mod-Gaussian convergence results.

Abstract

We consider the dilute Curie-Weiss model of size $N$, which is a generalization of the classical Curie-Weiss model where the dependency structure between the spins is not encoded by the complete graph but via the (directed) Erd\H{o}s-R\'enyi graph on $N$ vertices in which every edge appears independently with probability $p(N)$. In the high temperature with external magnetic field regime ($0<\beta<1,h\in\mathbb{R}$) we prove for $p^{3}N^{2}\to\infty$ sharp cumulant bounds for the magnetization for the annealed Gibbs measure implying a central limit theorem with rate, a moderate deviation principle, a concentration inequality, a normal approximation bound with Cram\'er correction and mod-Gaussian convergence.