Abstract
We consider the monomer–dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We characterise this limit as the expectation of the solution of a fixed point distributional equation and we give an explicit expression of the limiting pressure per particle.
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Roberts J.K.: Some properties of adsorbed films of oxygen on tungsten. Proc. R. Soc. Lond. A 152(876), 464–477 (1935)
Heilmann O.J., Lieb E.H.: Theory of monomer–dimer systems. Commun. Math. Phys. 25, 190–232 (1972)
Heilmann O.J., Lieb E.H.: Monomers and dimers. Phys. Rev. Lett. 24(25), 1412–1414 (1970)
Kasteleyn P.W.: The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27(12), 1209–1225 (1961)
Fisher M.E.: Statistical mechanics of dimers on a plane lattice. Phys. Rev. 124(6), 1664–1672 (1961)
Temperley H.N.V., Fisher M.E.: Dimer problem in statistical mechanics: an exact result. Philos. Mag. Ser. 8 6(68), 1061–1063 (1961)
Dembo A., Montanari A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2), 565–592 (2010)
Aldous D., Steele J.M.: The objective method: probabilistic combinatorial optimization and local weak convergence. Encycl. Math. Sci. 110, 1–72 (2004)
Bordenave C., Lelarge M., Salez J.: Matchings on infinite graphs. Probab. Theory Relat. Fields 157, 183–208 (2013)
Karp, R., Sipser, M.: Maximum matchings in sparse random graphs. In: Proceedings of 22nd Annual Symposium Foundation on Computer Science. IEEE Comput. Soc. Press, Silver Spring, pp. 364–375 (1981)
Salez J.: Weighted enumeration of spanning subgraphs in locally tree-like graphs. Random Struct. Algorithms 43(3), 377–397 (2013)
Lelarge, M.: A new approach to the orientation of random hypergraphs. In: Proceedings of 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (2012)
Zdeborová L., Mézard M.: The number of matchings in random graphs. J. Stat. Mech. 5, P05003 (2006)
Bayati, M., Nair, C.: A rigorous proof of the cavity method for counting matchings. In: Proceedings of 44th Annual Allerton Conference on Communication Control Computing (2006)
Lang S.: Complex Analysis, 4th ed. Springer, Berlin (1999)
Billingsley P.: Probability and Measure, 3rd ed. Wiley, London (1995)
Billingsley P.: Convergence of Probability Measures, 2nd ed. Wiley, London (1999)
Dembo A., Montanari A.: Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24(2), 137–211 (2010)
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Communicated by H. Spohn
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Alberici, D., Contucci, P. Solution of the Monomer–Dimer Model on Locally Tree-Like Graphs. Rigorous Results. Commun. Math. Phys. 331, 975–1003 (2014). https://doi.org/10.1007/s00220-014-2080-3
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DOI: https://doi.org/10.1007/s00220-014-2080-3