Decay of correlations in 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 transverse spin-spin correlations in classical spin systems. Our proof is based on the cluster expansion technique.

arXiv comment8 pages, 1 figure, title has changed, modifications in the proof of Theorem 2.1, published at JMP