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

作者：Bedini Andrea^*; Jacobsen Jesper Lykke

来源：Journal of Physics A-Mathematical and Theoretical, 2010, 43(38): 385001.

DOI：10.1088/1751-8113/43/38/385001

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 similar to exp(1.516 root N), a substantial improvement over the exponential running time similar to exp(0.245N) 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.

• 出版日期2010-9-24