Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs

TL;DR

This paper extends known probabilistic results of the Curie-Weiss model to a diluted ferromagnetic model on Erdős-Rényi graphs, showing convergence of spin configurations and deriving laws of large numbers and CLTs.

Contribution

It generalizes previous results on the Curie-Weiss model to diluted Erdős-Rényi graphs for the regime where Np→∞, including convergence in distribution and limit theorems.

Findings

Weak convergence of spin configuration vectors under the diluted model

Law of large numbers for two disjoint spin groups

Central limit theorem for the same groups

Abstract

This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model. We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erd\H{o}s-R\'enyi random graph. Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit. If $Np\longrightarrow\infty$ in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al. (2020) proved central limit theorems in probability. Here, we generalise these results for $\beta<1$ in the regime $Np\longrightarrow\infty$. We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations. This generalises various results from the Curie-Weiss model to the diluted model. As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins.