Kesten's incipient infinite cluster and quasi-multiplicativity of crossing probabilities

被引：2
作者：
Basu, Deepan [1 ]
Sapozhnikov, Artem [2 ]
机构：
[1] Max Planck Inst Math Sci, Inselstr 22, D-04103 Leipzig, Germany
[2] Univ Leipzig, Dept Math, Augustuspl 10, D-04109 Leipzig, Germany
来源：
ELECTRONIC COMMUNICATIONS IN PROBABILITY | 2017年 / 22卷
关键词：
incipient infinite cluster; percolation; criticality; quasi-multiplicativity; slab; PERCOLATION;
D O I：
10.1214/17-ECP56
中图分类号：
O21 [概率论与数理统计]; C8 [统计学];
学科分类号：
020208 ; 070103 ; 0714 ;
摘要：
In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph G. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten's incipient infinite cluster. We show that our assumptions are satisfied if G is a slab Z(2) x {0, ... , k}(d-2) (d >= 2, k >= 0). We also argue that the quasi-multiplicativity assumption should hold for G = Z(d) when d < 6, but not when d > 6.
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页数：12
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