A lower bound on the critical parameter of interlacement percolation in high dimension
A lower bound on the critical parameter of interlacement percolation in high dimension
复制标题
高维交错渗流临界参数下界
DOI：
10.1214/10-aop545
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发表时间：
2010
影响因子：
2
通讯作者：
A. Sznitman
中科院分区：
文献类型：
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作者：
A. Sznitman
我们研究了当d较大时，随机交织在$${\mathbb{Z}^d}$$上所留下的空集的分解性质.非负参数u控制$${\mathbb{Z}^d}$$上随机交错的密度。从Sznitman（Ann Math，2010）以及Sidoravicius和Sznitman（Comm Pure Appl Math 62（6）：831-858，2009）可知，存在非退化临界值u*，使得当u < u* 时，水平u处的空集发生渗透，并且当u > u* 时，不发生渗透。关于u* 知之甚少，然而，对于大d，$${\mathbb{Z}^d}$$上的随机交织应该表现出与（2d）-正则树上的随机交织的相似性，其中可以明确地计算对应的临界参数，参见特谢拉（Electron J Probab 14：1604-1627，2009）。本文证明了lim infd u*/ log d ≥ 1。这一下界与上述理论是一致的。
We investigate the percolative properties of the vacant set left by random interlacements on $${\mathbb{Z}^d}$$, when d is large. A non-negative parameter u controls the density of random interlacements on $${\mathbb{Z}^d}$$. It is known from Sznitman (Ann Math, 2010), and Sidoravicius and Sznitman (Comm Pure Appl Math 62(6):831–858, 2009), that there is a non-degenerate critical value u*, such that the vacant set at level u percolates when u < u*, and does not percolate when u > u*. Little is known about u*, however, random interlacements on $${\mathbb{Z}^d}$$, for large d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, where the corresponding critical parameter can be explicitly computed, see Teixeira (Electron J Probab 14:1604–1627, 2009). We show in this article that lim infd u*/ log d ≥ 1. This lower bound is in agreement with the above mentioned heuristics.