Solution of the monomer-dimer model on locally tree-like graphs. Rigorous results

TL;DR

This paper rigorously analyzes the monomer-dimer model on locally tree-like graphs, proving convergence of monomer density and providing explicit formulas for the thermodynamic limit.

Contribution

It introduces a novel rigorous framework for analyzing the monomer-dimer model on random graphs converging to trees, including explicit formulas for the limiting quantities.

Findings

Monomer density converges almost surely to an analytic function.

The limit is characterized by a fixed point distributional equation.

An explicit expression for the limiting pressure per particle is provided.

Abstract

We consider the monomer-dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We characterise this limit as the expectation of the solution of a fixed point distributional equation and we give an explicit expression for the limiting pressure per particle.