8,630 research outputs found

    A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

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    Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded. Version 3 shows that the worst-case running time is sub-exponential in the number of vertice

    Thin Fisher Zeroes

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    Biskup et al. [Phys. Rev. Lett. 84 (2000) 4794] have recently suggested that the loci of partition function zeroes can profitably be regarded as phase boundaries in the complex temperature or field planes. We obtain the Fisher zeroes for Ising and Potts models on non-planar (``thin'') regular random graphs using this approach, and note that the locus of Fisher zeroes on a Bethe lattice is identical to the corresponding random graph. Since the number of states appears as a parameter in the Potts solution the limiting locus of chromatic zeroes is also accessible.Comment: 10 pages, 4 figure

    Thin Fisher zeros

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    Various authors have suggested that the loci of partition function zeros can profitably be regarded as phase boundaries in the complex temperature or field planes. We obtain the Fisher zeros for Ising and Potts models on non-planar ('thin') regular random graphs using this approach, and note that the locus of Fisher zeros on a Bethe lattice is identical to the corresponding random graph. Since the number of states q appears as a parameter in the Potts solution the limiting locus of chromatic zeros is also accessible
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