14-74 P. Butera, P. Federbush, M. Pernici

Positivity of the virial coefficients in lattice dimer models (34K, LaTeX) Nov 2, 14

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Abstract. Using a simple relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer-dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have also observed that all virial coefficients so far known for this system are positive also on various infinite regular lattices of a different structure. We are thus led to conjecture that all of them are always positive. These considerations are generalized to the study of related bounds on finite graphs bearing some similarity to infinite regular lattice models, namely regular biconnected graphs and finite grids. The validity of the bounds Delta^k (ln(i! N(i))) < 0 for k > 1, where N(i) is the number of configurations of i dimers on the graph and Delta is the finite difference operator, is shown to correspond to the positivity of the virial coefficients. An exhaustive survey of some classes of regular biconnected graphs with a not too large number v of vertices shows that there are only few violations of these bounds. We conjecture that the frequency of the violations vanishes as v goes to infinity. These bounds are valid also for square and triangular N x N grids with N < 20 or N < 19 respectively, and with open boundary conditions, giving some support to the conjecture on the positivity of the virial coefficients.

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