Validations of the Asymptotic Matching Conjectures

Details Validations of the Asymptotic Matching Conjectures Validations of the Asymptotic Matching Conjectures

2006-02-28

arxiv.org/abs/math/0603001v3

Mathematics

Combinatorics

25 pages, 9 figures

Scientific paper

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 $p\in [0,1]$, 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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