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

Abstract

Let lambda(d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z(d), where p is an element of [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). Converted to a power series in p, our " formal" expansion shows remarkable validity in low dimensions, d = 1, 2, 3, in which dimensions we give some numerical studies.