Abstract:Consider all colorings of a finite box in a multidimensional grid with a given number of colors subject to given local constraints. We outline the most recent theory for the computation of the exponential growth rate of the number of such colorings as a function of the dimensions of the box. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 9 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%. … Show more
“…See for example [1,2,4,5,6,12,14,15,16,17,19,20,21,23,24,25,26,28,30]. Let G = (V, E) be an undirected graph with vertices V and edges E. G can be a finite or infinite graph.…”
In this paper we review the asymptotic matching conjectures for rregular bipartite graphs, and their connections in estimating the monomerdimer 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.2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20 Keywords and phrases: Matching and asymptotic growth of average matchings for r-regular bipartite graphs, monomer-dimer partitions and entropies.
“…See for example [1,2,4,5,6,12,14,15,16,17,19,20,21,23,24,25,26,28,30]. Let G = (V, E) be an undirected graph with vertices V and edges E. G can be a finite or infinite graph.…”
In this paper we review the asymptotic matching conjectures for rregular bipartite graphs, and their connections in estimating the monomerdimer 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.2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20 Keywords and phrases: Matching and asymptotic growth of average matchings for r-regular bipartite graphs, monomer-dimer partitions and entropies.
“…The middle term is obtained by considering the eigenvalues of A 3,5 , the 10 by 10 matrix whose row and columns are indexed by lexicographically ordered 3-element subsets of [5]. (See [9][10][11].) From the previous section we deduce h(m) = log ρ(m) 2m .…”
Section: The Transfer Matrix Methods For Counting Perfect Matchings Inmentioning
A connected planar graph is called m-generalized fullerene if two of its faces are mgons and all other faces are pentagons and hexagons. In this paper we first determine some structural properties of m-generalized fullerenes and then use them to obtain new results on the enumerative aspects of perfect matchings in such graphs. We provide both upper and lower bounds on the number of perfect matchings in m-generalized fullerene graphs and state exact results in some special cases.
“…Hammersley proved in (19) that the number of p-density configurations on the cube of volume n in d dimensions is of order ȕ(d, p) n for some function ȕ. He spent much effort on obtaining bounds for ȕ, but, even today in two dimensions, our knowledge is very limited (see, for example, Friedland & Peled 2005).…”
Section: Self-avoiding Walks and The Monomer-dimer Problemmentioning
John Hammersley was a pioneer among mathematicians, who defied classification as pure or applied; when introduced to guests at Trinity College, Oxford, he would say he did ‘difficult sums’. He believed passionately in the importance of mathematics with strong links to real–life situations and in a system of mathematical education in which the solution of problems takes precedence over the generation of theory. He will be remembered for his work on percolation theory, subadditive stochastic processes, self–avoiding walks and Monte Carlo methods, and, by those who knew him, for his intellectual integrity and his ability to inspire and to challenge. Quite apart from his extensive research achievements, for which he earned a reputation as an outstanding problem–solver, he was a leader in the movement of the 1950s and 1960s to rethink the content of school mathematics syllabuses.
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