bibkey: butera2015virial authors: Paolo Butera; Paul Federbush; Mario Pernici year: 2015 title: “Positivity of the virial coefficients in lattice dimer models and upper bounds on the number of matchings on graphs” doi: 10.1016/j.physa.2015.05.106 url: https://arxiv.org/abs/1502.06734v2 claim: “For a finite regular graph with N(i) the number of configurations of i dimers (i-edge matchings) and nu the matching number, the paper argues that the bounds Delta^k ln(i! N(i)) <= 0 for k = 2, …, nu and i = 0, …, nu - k (Eq. (1), with Delta the forward difference) correspond to the positivity of the virial coefficients, reports tests on lattice graphs and on regular biconnected graphs, and asks whether these bounds for k <= 4 always hold for regular biconnected graphs.” strata_touched:

• D5/S3/StatisticalMechanics/DimerVirialFourthDifferenceRefutation license: citation-only triage: anchor

Butera–Federbush–Pernici, virial positivity and matchings on graphs

P. Butera, P. Federbush, M. Pernici, Physica A 437 (2015) 278–294, arXiv:1502.06734v2 (cond-mat.stat-mech). Quotations are from the arXiv v2 source.

Abstract:

The validity of the bounds ${\Delta }^k\ln (i!N(i))\le 0$ for $k \ge 2$,where$N(i)$isthenumberofconfigurationsof$i$ dimers on the graph and $\Delta$ is the forward difference operator, is shown to correspond to the positivity of the virial coefficients.

Introduction, before Eq. (1):

Using the definition of the graph dimer entropy[\onlinecite{bfppos}], we argue that for a finite regular graph the bounds which correspond to the positivity of the virial coefficients $m_k$ for infinite regular lattices are

followed by Eq. (1), ${\Delta }^k\ln (i!N(i))\le 0$, and

with $k=2,\dots ,\nu$ and $i=0,\dots ,\nu -k$, where $\nu$ is the matching number of $G$, i.e. the maximum number of pairwise disjoint edges of $G$.

Section IV B, after the tests on finite lattices with open boundary conditions:

For all the graphs examined in this section, Eq. (\ref{Delta0}) is satisfied for $k\le 4$. It would be interesting to know whether these bounds, Eq. (\ref{Delta0}) for $k\le 4$, are always satisfied for regular biconnected graphs.

The Conclusions note that for $k=2$ the bounds follow from the Heilmann–Lieb inequality. Sections IV C and IV D test Eq. (1) on regular biconnected bipartite and non-bipartite graphs and report violations only for $k\ge 5$.

The encoding reads $N(i)$ as the number of $i$-edge matchings of $K_n$ all of whose edges are edges of $G$, $\nu$ as the supremum of the $i$ with $N(i)>0$, $\Delta$ as fwdDiff 1, and “regular biconnected” as: every vertex has the same degree, $G$ is connected, and $G-v$ is connected for every vertex $v$.

Verified locator

• DOI: https://doi.org/10.1016/j.physa.2015.05.106 (Physica A 437 (2015) 278–294).
• URL: https://arxiv.org/abs/1502.06734v2 (the latest version, 2015-05-20; source virial8j_c3b_arXiv.tex, md5 26db1bd9f13d1a867af773dc8058ff37): abstract (l. 56–60), Heilmann–Lieb sentence (l. 105–107), Eq. (1) (l. 120–128), the question (l. 1010–1013), tests on bipartite and non-bipartite graphs (l. 1017–1148), Conclusions (l. 1461–1480).