On long range percolation with heavy tails. (English) Zbl 1060.60093

Friedli, Sacha; de Lima, Bernardo Nunes Borges; Sidoravicius, Vladas

Electron. Commun. Probab. 9, 175-177 (2004).

Summary: Consider independent long range percolation on \(\mathbb Z^d\), where edges of length \(n\) are open with probability \(p_n\). We show that if \(\lim\sup_{n\to\infty}p_n>0,\) then there exists an integer \(N\) such that \(P_N(0\leftrightarrow \infty)>0\), where \(P_N\) is the truncated measure obtained by taking \(p_{N,n}=p_n\) for \(n \leq N\) and \(p_{N,n}=0\) for all \(n> N\).

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B43	Percolation

Keywords:

long range percolation; truncation; slab percolation