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

Paul Federbush,Shmuel Friedland

doi:10.1007/s10955-011-0170-6

Paul Federbush, Shmuel Friedland

Open Access

https://doi.org/10.1007/s10955-011-0170-6

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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).

Highlights

The first aim of this paper is to discuss an asymptotic expansion of the monomer-dimer p-entropy, denoted by λd (p), where p ∈ [0, 1] is the density of dimers, on the integer d-dimensional lattice Zd
The study of the existence of this entropy, some of its properties and its estimates, was initiated in a series of papers by Hammersley and his collaborators [19, 20, 21, 22]. It was shown by Minc [24] that the d-dimensional dimer entropy λd (1) satisfies
The lower bound is implied by the proof of the van der Waerden conjecture [5, 6], or its weak form [15]

Summary

Introduction

The first aim of this paper is to discuss an asymptotic expansion of the monomer-dimer p-entropy, denoted by λd (p), where p ∈ [0, 1] is the density of dimers, on the integer d-dimensional lattice Zd. The study of the existence of this entropy, some of its properties and its estimates, was initiated in a series of papers by Hammersley and his collaborators [19, 20, 21, 22]. The last upper bound follows from a sharp form of the Stirling formula for (2d)! In a series of papers [7] – [11], the first author studied a possible asymptotic expansion of λd. This is derived assuming that an argument employing a formal cluster expansion could be made rigorous. Our expansion was developed as an asymptotic expansion in 1/d it has surprising validity for small d

Preliminary Results

The Asymptotic Expansion

The LAMC and Other Conclusions