Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters

Abstract

We study the distribution of finite clusters in slightly supercritical (p↓pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition (pc<p2→2) and K denotes the cluster of the origin then there exists δ>0 such that Pp(n≤|K|<∞)≍n−1/2exp[−Θ(|p−pc|2n)] and Pp(r≤Rad(K)<∞)≍r−1exp[−Θ(|p−pc|r)] for every p∈(pc−δ,pc+δ) and n,r≥1, where all implicit constants depend only on G. We deduce in particular that the critical exponents γ′ and Δ′ describing the rate of growth of the moments of a finite cluster as p↓pc take their mean-field values of 1 and 2 respectively. These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on ℤd even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.