Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model
Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model
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DOI：
10.1214/aop/1020107775
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发表时间：
2000-08
影响因子：
2.3
通讯作者：
O. Haggstrom;J. Jonasson;R. Lyons
中科院分区：
文献类型：
--
作者：
O. Haggstrom;J. Jonasson;R. Lyons
随机簇模型是一种非独立的渗流模型，在Ising和Potts模型的研究中有应用。本文对簇参数q > 1的非顺从图上的随机簇模型得到了几个新的结果。其中，主要的是自由随机簇测度在临界值处不存在渗流，以及具有正则对偶的平面正则图的例子，其中p free c（q）> p wired u（q），q足够大。后者如下考虑等周常数，我们给第一个非平凡的明确计算等常数。这样的考虑也被用来证明非鲁棒相变的Potts模型上的不服从规则的图形。
The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter q > 1. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value and examples of planar regular graphs with regular dual where p free c (q) > p wired u (q) for q large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove nonrobust phase transition for the Potts model on nonamenable regular graphs.