Phase transition in loop percolation

• 成果类型: Article
• 署名作者: Chang, Yinshan; Sapozhnikov, Artem
• 署名单位: Max Planck Society
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0624-x
• 发表日期: 2016
• 页码: 979-1025
• 关键词: random-walk

摘要

We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13-14):643-646, 2012, Ill J Math 57(2):525-558, 2013). It is a model with long range correlations with two parameters and . The non-negative parameter measures the amount of loops, and plays the role of killing on vertices penalizing () or favoring () appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525-558, 2013) that for any fixed and large enough , there exists an infinite cluster in the loop percolation on . In the present article, we show a non-trivial phase transition on the integer lattice () for . More precisely, we show that there is no loop percolation for and small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of , namely, for and any sub-critical value of , the probability of one-arm event decays at most polynomially. For , we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity .