Percolation Connectivity in the Highly Supercritical Regime

#### G.A. Braga, A. Procacci, R. Sanchis, Benedetto Scoppola

2004, v.10, №4, 607-628

ABSTRACT

We prove that the two point finite connectivity function $\t^{\rm f}(x,y)$ in the $d$-dimensional Bernoulli bond percolation is analytic in $p$ around $1$. We also provide upper and lower bounds for this function in the case $d\ge 3$ and near $p=1$. The gap between lower bound and upper bound is sufficiently narrow to conclude that the rate of exponential decay, i.e. the inverse correlation length $m(p)$, is, for $p$ sufficiently near to $1$ and for $x-y$ in the coordinate axis directions, of the form $m(p)=2(d-1)|\ln (1-p)|+O(1-p)$, as expected by intuition based on low temperature expansion arguments.

Keywords: supercritical percolation,finite connectivity