P. Butera, P. Federbush, M. Pernici
Positivity of the virial coefficients in lattice dimer models
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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.