13-31 P. Butera, P. Federbush, M. Pernici

Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices (55K, LaTeX) Apr 4, 13

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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 dimer-density p-dependent entropy of a monomer-dimer system involving a sum of 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 hypercubic lattices of general dimensionality d and from 6 to 24 for the lattices of dimensionalities 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 hypercubic 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 the paper self-contained by including a review of the earlier works.

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