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.
Similar content being viewed by others
References
Abért, M., Csikvári, P., Frenkel, P.E., Kun, G.: Matchings in Benjamini-Schramm convergent graph sequences. Trans. Am. Math. Soc. arXiv:1405.3271
Abért, M., Hubai, T.: Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs. arXiv:1201.3861, to appear in Combinatorica
Abért, M., Thom, A., Virág, B.: Benjamini-Schramm convergence and pointwise convergence of the spectral measure. www.renyi.hu/~abert
Alm, S.E.: Upper bounds for the connective constant of self-avoiding walks. Comb. Probab. Comput. 2, 115–136 (1993)
Baxter, R.J.: Dimers on a rectangular lattice. J. Math. Phys. 9, 650–654 (1968)
Butera, P., Federbush, P., Pernici, M.: Higher order expansion for the entropy of a dimer or a monomer-dimer system on \(d\)- dimensional lattices. Phys. Rev. E 87, 062113 (2013)
Butera, P., Pernici, M.: Yang-Lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality \(2\le d\le 7\). Phys. Rev. E 86, 011104 (2012)
Csikvári, P., Frenkel, P.E.: Benjamini-Schramm continuity of root moments of graph polynomials. arXiv:1204.0463
Darroch, J.N.: On the distribution of the number of successes in independent trials. Ann. Math. Stat. 35, 1317–1321 (1964)
Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt{2}}\). Ann. Math. 175(3), 1653–1665 (2012)
Elek, G., Lippner, G.: Borel oracles. An analytical approach to constant-time algorithms. Proc. Am. Math. Soc. 138(8), 2939–2947 (2010)
Fisher, M.E.: Statistical mechanics of dimers on a plane lattice. Phys. Rev. 124, 1664–1672 (1961)
Friedland, S., Gurvits, L.: Lower bounds for partial matchings in regular bipartite graphs and applications to the monomer-dimer entropy. Comb. Probab. Comput. 17, 347–361 (2008)
Friedland, S., Peled, U.N.: Theory of computation of multidimensional entropy with an application to the monomer-dimer problem. Adv. Appl. Math. 34, 486–522 (2005)
Gamarnik, D., Katz, D.: Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys. 137, 205–232 (2009)
Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)
Gurvits, L.: Unleashing the power of Schrijver’s permanental inequality with the help of the Bethe approximation. arXiv:1106.2844v11
Hara, T., Slade, G., Sokal, A.D.: New lower bounds on the self-avoiding-walk connective constant. J. Stat. Phys. 72, 479–517 (1993)
Hammersley, J.M.: Existence theorems and Monte Carlo methods for the monomer-dimer problem. In: David, (ed.) Research Papers in Statistics: Festschrift for J. Neyman, pp. 125–146. Wiley, London (1966)
Hammersley, J.M.: An improved lower bound for the multidimensional dimer problem. Proc. Camb. Philos. Soc. 64, 455–463 (1966)
Hammersley, J.M., Menon, V.: A lower bound for the monomer-dimer problem. J. Inst. Math. Appl. 6, 341–364 (1970)
Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. Commun. Math. Phys. 25, 190–232 (1972)
Huo, Y., Liang, H., Liu, S.Q., Bai, F.: Computing the monomer-dimer systems through matrix permanent. Phys. Rev. E 77, 016706 (2008)
Kasteleyn, P.W.: The statistics of dimers on a lattice, I: the number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)
Ku, C.Y., Chen, W.: An analogue of the Gallai-Edmonds structure theorem for non-zero roots of the matching polynomial. J. Comb. Theory Ser. B 100, 119–127 (2010)
Nguyen, H.N., Onak, K.: Constant-time approximation algorithms via local improvements. In: 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 327–336 (2008)
McKay, B.D.: The expected eigenvalue distribution of a large regular graph. Linear Algebr. Appl. 40, 203–216 (1981)
Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics-an exact result. Philos. Mag. 6, 1061–1063 (1961)
Thom, A.: Sofic groups and diophantine approximation. Commun. Pure Appl. Math. LXI, 1155–1171 (2008)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-015-1309-7