k-independent percolation on trees

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Mathieu, Pierre²  Temmel, Christoph¹ 

[1] Graz Univ Technol, Inst Math Strukturtheorie 5030, A-8010 Graz, Austria

[2] Univ Aix Marseille 1, LATP, CMI, UMR CNRS 6632, F-13453 Marseille 13, France

关键词: k-independent;    k-dependent;    Tree percolation;    Critical value;    Percolation kernel;    Second moment method;    Shearer's measure;   

DOI  :  10.1016/j.spa.2011.10.014

来源: Elsevier

【 摘 要 】

Consider the class of k-independent bond or site percolations with parameter p on a tree T. We derive tight bounds on p for both almost sure percolation and almost sure nonpercolation. The bounds are continuous functions of k and the branching number of T. This extends previous results by Lyons for the independent case (k = 0) and by Balister & Bollobas for 1-independent bond percolations. Central to our argumentation are moment method bounds a la Lyons supplemented by explicit percolation models a la Balister & Bollobas. An indispensable tool is the mini mality and explicit construction of Shearer's measure on the k-fuzz of Z. (C) 2011 Published by Elsevier B.V.

【 授权许可】

Free