Cluster-size decay in supercritical long-range percolation

We study the cluster-size distribution of supercritical long-range percolation on Z^d, where two vertices x, y ϵ Z^d are connected by an edge with probability p(‖x − y‖):= p min(1, β‖x − y‖)^−dα for parameters p ϵ (0, 1], α > 1, and β > 0. We show that when α > 1 + 1/d, and either β or p is sufficiently large, the probability that the origin is in a finite cluster of size at least k decays as exp( − Θ(k^(d−1)/d)). This corresponds to classical results for nearest-neighbor Bernoulli percolation on Z^d, but is in contrast to long-range percolation with α < 1 + 1/d, when the exponent of the stretched exponential decay changes to 2 − α. This result, together with our accompanying paper, establishes the phase diagram of long-range percolation with respect to cluster-size decay. Our proofs rely on combinatorial methods that show that large delocalized components are unlikely to occur. As a side result we determine the asymptotic growth of the second-largest connected component when the graph is restricted to a finite box.