渗透理论中的唯一性与非唯一性

IF 1.3 Q2 STATISTICS & PROBABILITY
O. Haggstrom, J. Jonasson
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引用次数: 80

摘要

本文是对${\mathbb{Z}}^d$上的无限簇的唯一性与非唯一性问题的最新介绍,更一般地说,在传递图上。对于${\mathbb{Z}}^d$上的iid渗透,无限簇的唯一性是一个经典结果,而在某些其他传递图上,唯一性可能失效。在这种情况下,图的关键性质是可修改性和不可修改性。同样的问题也适用于某些依赖的渗透模型——最突出的是Fortuin- Kasteleyn随机聚类模型——以及标准连通性概念被纠缠或刚性取代的情况。还考虑了渗流过程耦合中的所谓同时唯一性。有些主要的结果得到了详细的证明,有些则只是简略地证明,有些则略去了。讨论了几个悬而未决的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Uniqueness and Non-uniqueness in Percolation Theory
This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d$ and, more generally, on transitive graphs. For iid percolation on ${\mathbb{Z}}^d$, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
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