A note on the monomer-dimer model

Abstract

We consider the monomer-dimer model, whose realisations are spanning sub-graphs of a given graph such that every vertex has degree zero or one. The measure depends on a parameter, the monomer activity, which rewards the total number of monomers. We consider general correlation functions including monomer-monomer correlations and dimer-dimer covariances. We show that these correlations decay exponentially fast with the distance if the monomer activity is strictly positive. Our result improves a previous upper bound from van den Berg and is of interest due to its relation to truncated spin-spin correlations in classical spin systems. Our proof is based on the cluster expansion technique.