An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

TL;DR

This paper develops asymptotic expansions and recursive inequalities to bound the monomer-dimer entropy on high-dimensional lattices, providing new insights into its behavior and confirming a key conjecture.

Contribution

It introduces bounds for monomer-dimer entropy using recursive relations and confirms the lower asymptotic matching conjecture.

Findings

Derived upper and lower bounds for monomer-dimer entropy.

Computed the first three terms of the asymptotic expansion in powers of 1/d.

Confirmed the lower asymptotic matching conjecture for the entropy.

Abstract

Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).