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On the Validations of the Asymptotic Matching Conjectures

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Abstract

In this paper we review the asymptotic matching conjectures for r-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p∈[0,1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA

    S. Friedland & E. Krop

  2. Berlin Mathematical School, Berlin, Germany

    S. Friedland

  3. Department of Physics, AlbaNova University Center, KTH, 106 91, Stockholm, Sweden

    P. H. Lundow

  4. Department of Mathematics and Mathematical Statistics, Umeå University, 901 87, Umeå, Sweden

    K. Markström

Authors
  1. S. Friedland
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  2. E. Krop
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  3. P. H. Lundow
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  4. K. Markström
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Correspondence to S. Friedland.

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Friedland, S., Krop, E., Lundow, P.H. et al. On the Validations of the Asymptotic Matching Conjectures. J Stat Phys 133, 513–533 (2008). https://doi.org/10.1007/s10955-008-9550-y

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  • DOI: https://doi.org/10.1007/s10955-008-9550-y

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