计算任意图精确Potts模型配分函数的树分解转移矩阵及其在平面图着色中的应用
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
摘要：
结合树分解和转移矩阵技术提供了一个非常通用的算法来计算定义在任意图上的统计模型的精确配分函数。该算法是特别有效的平面图的情况下。我们说明它通过计算Potts模型配分函数和色多项式（适当的顶点着色使用Q颜色的数量）的随机平面图的大样本高达N=100顶点。在后一种情况下，我们的算法产生一个次指数平均运行时间~ exp（1.516 sqrt（N）），一个显着的改善指数运行时间~ exp（0.245 N）提供了迄今为止最知名的算法。我们研究了随机平面图的色根的统计在一定程度上的细节，比较的结果与有限块的一个规则格。
英文摘要：
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.
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