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

journal contribution

posted on 2011-05-26, 00:00 authored by Paul Federbush, Shmuel FriedlandLet 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.

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## Publisher Statement

Post print version of article may differ from published version. The original publication is available at www.springerlink.com; DOI: 10.1007/s10955-011-0170-6.## Publisher

Springer Verlag## Language

- en_US

## issn

0022-4715## Issue date

2011-04-01## Usage metrics

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