Central limit theorem in disordered Monomer‐Dimer model

Abstract

AbstractWe 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 ‐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 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.