Finite Connections for Supercritical Bernoulli Bond Percolation in 2D

#### M. Campanino, D. Ioffe, O. Louidor

2010, v.16, №2, 225-266

ABSTRACT

Two vertices x and y are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics of finite connections for super-critical Bernoulli bond percolation on $Z^2$. These asymptotics are based on a detailed fluctuation analysis of long finite super-critical clusters or, more precisely, of dual open (sub-critical) loops which surround such clusters.

Keywords: Bernoulli bond percolation,random walk representation,interacting random walks,Ornstein - Zernike decay of correlations,local limit theorems