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

TL;DR

This paper extends the known higher-order series expansions for the entropy of dimer and monomer-dimer systems on d-dimensional lattices, providing more accurate estimates and proposing a conjecture about the positivity of expansion coefficients.

Contribution

It significantly extends the number of known terms in the entropy expansions for hypercubic lattices, enabling better numerical estimates and proposing a new conjecture about coefficient positivity.

Findings

Extended the series to 10 terms for dimer entropy in d dimensions.

Extended the series to 20-24 terms for monomer-dimer entropy, improving numerical estimates.

Proposed the conjecture that all expansion coefficients are positive.

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.