Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices

Abstract

Recently an expansion as a power series in 1/d has been presented for the specific entropy of a complete dimer covering of a d-dimensional hypercubic lattice. This paper extends from 3 to 10 the number of terms known in the series. Likewise an expansion for the entropy, dependent on the dimer-density p, of a monomer-dimer system, involving a sum sum_k a_k(d) p^k, has been recently offered. We herein extend the number of the known expansion coefficients from 6 to 20 for the hyper-cubic lattices of general dimension d and from 6 to 24 for the hyper-cubic lattices of dimensions d < 5 . We show that this extension can lead to accurate numerical estimates of the p-dependent entropy for lattices with dimension d > 2. The computations of this paper have led us to make the following marvelous conjecture: "In the case of the hyper-cubic lattices, all the expansion coefficients, a_k(d), are positive"! This paper results from a simple melding of two disparate research programs: one computing to high orders the Mayer series coefficients of a dimer gas, the other studying the development of entropy from these coefficients. An effort is made to make this paper self-contained by including a review of the earlier works.

17 pages, no figures

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• The high-temperature expansions of the higher susceptibilities for the Ising model in general dimension d
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Cited by in corpus (7)

• Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy
• A positivity property of the dimer entropy of graphs
• Power series expansions for the planar monomer-dimer problem
• Errata and Addenda to Mathematical Constants
• Four Amazing Positivities with Dimers/i-Matchings
• Random Regular Bipartite Graphs Satisfy Weak Virial Positivity, for a Large Range of the Parameters
• Matchings in vertex-transitive bipartite graphs