Central Limit Theorem in Disordered Monomer-Dimer Model

TL;DR

This paper proves a Gaussian central limit theorem for the free energy in the disordered monomer-dimer model on finite graphs, extending previous results to more general graph classes and weaker moment conditions.

Contribution

It generalizes the CLT for the disordered monomer-dimer model to broader graph classes and weaker weight distribution assumptions, with a new decay of correlation approach.

Findings

CLT for free energy under finite fourth moment assumption

CLT under nearly optimal finite (2+ε)-moment assumption for graphs with subexponential growth

CLT for the Gibbs average of the number of dimers on certain graphs

Abstract

We consider the disordered monomer-dimer model on general finite graphs with bounded degrees. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the associated Gibbs measure with a rate of convergence. The central limit theorem continues to hold under a nearly optimal finite $(2+\epsilon)$-moment assumption on the weight distributions if the underlying graphs are further assumed to have a uniformly subexponential volume growth. This generalizes a recent result by Dey and Krishnan (arXiv:2109.12716) who showed a Gaussian central limit theorem in the disordered monomer-dimer model on cylinder graphs. Our proof relies on the idea that the disordered monomer-dimer model exhibits a decay of correlation with high probability. We also establish a central limit theorem for the Gibbs average of the number of dimers where the underlying graph has subexponential volume growth and the edge weights are Gaussians.