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
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
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
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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