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Matching Measure, Benjamini–Schramm Convergence and the Monomer–Dimer Free Energy

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Abstract

We define the matching measure of a lattice L as the spectral measure of the tree of self-avoiding walks in L. We connect this invariant to the monomer–dimer partition function of a sequence of finite graphs converging to L. This allows us to express the monomer–dimer free energy of L in terms of the matching measure. Exploiting an analytic advantage of the matching measure over the Mayer series then leads to new, rigorous bounds on the monomer–dimer free energies of various Euclidean lattices. While our estimates use only the computational data given in previous papers, they improve the known bounds significantly.

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Acknowledgments

The authors are very grateful to Paolo Butera and Paul Federbush for many helpful discussions. All authors are partially supported by MTA Rényi “Lendület” Groups and Graphs Research Group. The second author is also partially supported by the National Science Foundation under Grant No. DMS-1500219. We are very grateful to the anonymous referee for helpful comments and suggestions.

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Correspondence to Péter Csikvári.

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Abért, M., Csikvári, P. & Hubai, T. Matching Measure, Benjamini–Schramm Convergence and the Monomer–Dimer Free Energy. J Stat Phys 161, 16–34 (2015). https://doi.org/10.1007/s10955-015-1309-7

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  • DOI: https://doi.org/10.1007/s10955-015-1309-7

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