A positivity property of the dimer entropy of graphs

Abstract

The entropy of a monomer-dimer system on an infinite bipartite lattice can be written as a mean-field part plus a series expansion in the dimer density. In a previous paper it has been conjectured that all coefficients of this series are positive. Analogously on a connected regular graph with $v$ vertices, the "entropy" of the graph $\ln N(i)/v$, where $N(i)$ is the number of ways of setting down $i$ dimers on the graph, can be written as a part depending only on the number of the dimer configurations over the completed graph plus a Newton series in the dimer density on the graph. In this paper, we investigate for which connected regular graphs all the coefficients of the Newton series are positive (for short, these graphs will be called positive). In the class of connected regular bipartite graphs, up to $v=20$, the only non positive graphs have vertices of degree $3$. From $v=14$ to $v=30$, the frequency of the positivity violations in the $3$-regular graphs decreases with increasing $v$. In the case of connected $4$-regular bipartite graphs, the first violations occur in two out of the $2806490$ graphs with $v=22$. We conjecture that for each degree $r$ the frequency of the violations, in the class of the $r-$regular bipartite graphs, goes to zero as $v$ tends to infinity. This graph-positivity property can be extended to non-regular or non-bipartite graphs. We have examined a large number of rectangular grids of size $N_x\times N_y$ both with open and periodic boundary conditions. We have observed positivity violations only for $min(N_x,N_y)=3$ or $4$.

arXiv comment13 pages, 1 figure

References in corpus (5)

• Unleashing the power of Schrijver's permanental inequality with the help of the Bethe Approximation
• Yang-Lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 <= d <= 7
• Higher order expansions for the entropy of a dimer or a monomer-dimer system on d-dimensional lattices
• An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem
• Sums of permanental minors using Grassmann algebra

Cited by in corpus (5)

• A Near Proof of Weak Graph Positivity, A New Property of Random Regular Graphs
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• The Genius Conjectures (via Bell Polynomials)
• Random Regular Bipartite Graphs Satisfy Weak Virial Positivity, for a Large Range of the Parameters
• Four Amazing Positivities with Dimers/i-Matchings