Decay of correlations in the monomer-dimer model

TL;DR

This paper proves that correlations in the monomer-dimer model decay exponentially with distance when monomer activity is positive, using cluster expansion techniques, improving previous bounds and relating to classical spin systems.

Contribution

It establishes exponential decay of correlations in the monomer-dimer model for positive monomer activity, refining earlier bounds and connecting to spin system correlations.

Findings

Correlations decay exponentially with distance at positive monomer activity

Improves previous upper bounds on correlation decay

Links monomer-dimer correlations to classical spin system behaviors

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.