3,109 research outputs found

    Ground states of two-dimensional ±\pmJ Edwards-Anderson spin glasses

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    We present an exact algorithm for finding all the ground states of the two-dimensional Edwards-Anderson ±J\pm J spin glass and characterize its performance. We investigate how the ground states change with increasing system size and and with increasing antiferromagnetic bond ratio xx. We find that that some system properties have very large and strongly non-Gaussian variations between realizations.Comment: 15 pages, 21 figures, 2 tables, uses revtex4 macro

    Factors of IID on Trees

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    Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur

    Characterizations of Mersenne and 2-rooted primes

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    We give several characterizations of Mersenne primes (Theorem 1.1) and of primes for which 2 is a primitive root (Theorem 1.2). These characterizations involve group algebras, circulant matrices, binomial coefficients, and bipartite graphs.Comment: 19 pages, final version, to appear in Finite Fields and their Application

    Approximating the monomer-dimer constants through matrix permanent

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    The monomer-dimer model is fundamental in statistical mechanics. However, it is #P-complete in computation, even for two dimensional problems. A formulation in matrix permanent for the partition function of the monomer-dimer model is proposed in this paper, by transforming the number of all matchings of a bipartite graph into the number of perfect matchings of an extended bipartite graph, which can be given by a matrix permanent. Sequential importance sampling algorithm is applied to compute the permanents. For two-dimensional lattice with periodic condition, we obtain 0.6627±0.0002 0.6627\pm0.0002, where the exact value is h2=0.662798972834h_2=0.662798972834. For three-dimensional lattice with periodic condition, our numerical result is 0.7847±0.0014 0.7847\pm0.0014, {which agrees with the best known bound 0.7653≤h3≤0.78620.7653 \leq h_3 \leq 0.7862.}Comment: 6 pages, 2 figure
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