Central limit theorem in disordered Monomer-Dimer model

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 + & varepsilon; ) $$ \left(2+\epsilon \right) $$ -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.