Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs

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 observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjecture that the virial expansion coefficients m k are always positive.These considerations can be extended to the study of related bounds on finite graphs generalizing the infinite regular lattices, namely the finite grids and the regular biconnected graphs. The validity of the bounds ∆ k ln(i!N (i)) ≤ 0 for k ≥ 2, where N (i) is the number of configurations of i dimers on the graph and ∆ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients.Our tests on many finite lattice graphs indicate that on large lattices these bounds are satisfied, giving support to the conjecture on the positivity of the virial coefficients. Moreover, in an exhaustive survey of some classes of regular biconnected graphs with a not too large number v of vertices, we observe only few violations of these bounds. We conjecture that the frequency of the violations vanishes as v → ∞.Using an inequality by Heilmann and Lieb, we find rigorous upper bounds on N (i) valid for arbitrary graphs and for regular graphs. The similarity between this inequality and the one conjectured above suggests that one study the stricter inequality m k ≥ 1 2k for the virial coefficients, which is valid for all the known coefficients of the infinite regular lattice models.

A note on Lee–Yang zeros in the negative half-plane

“…a finite number of negative) B j were for systems for which all the L-Y zeros lie on the negative z-axis. This behavior fits in with the conjecture by Federbush, et al, that all the B j for the monomer-dimer system on regular lattices are positive [37], since there the L-Y zeros are indeed on the negative z-axis [9]. Systems with strictly negative L-Y zeros do not have any phase transition, but this does not rule out the possibility that a system with a phase transition has almost all B j ≥ 0.…”

Random Regular Bipartite Graphs Satisfy Weak Virial Positivity, for a Large Range of the Parameters

“…Let m i be the number of imatchings. In [1], Butera, Pernici, and I introduced the quantity u(i) in eq (1) therein,…”

“…The term Virial comes from the Virial expansion in statistical mechanics. Equation (I.1) above corresponds, roughly speaking, to the positivity of the coefficients in the Virial Expansion for infinite regular lattices, [1]. For hyper-cubical lattices it is shown in [1] that the first 20 coefficients in the Virial expansion are positive in dimensions d ≤ 10 !…”

“…Equation (I.1) above corresponds, roughly speaking, to the positivity of the coefficients in the Virial Expansion for infinite regular lattices, [1]. For hyper-cubical lattices it is shown in [1] that the first 20 coefficients in the Virial expansion are positive in dimensions d ≤ 10 ! Our treatment of Virial Positivity follows slavishly the treatment of Graph Positivity in [3].…”

Four Amazing Positivities with Dimers/i-Matchings

“…In each section we italicize the conjecture in it towards which the effort is centered. The general references are for I, [1], for II, [3], for III, [2] and for IV, [3]. Additional references are separately noted.…”