Cluster-size decay in supercritical long-range percolation

Abstract

We study the cluster-size distribution of supercritical long-range percolation on ${\mathbb{Z}}^d$, where two vertices $x,y\in {\mathbb{Z}}^d$ are connected by an edge with probability $\mathrm{p}(\Vert x-y\Vert ):=pmin{(1,\mathit{\beta }\Vert x-y\Vert )}^{-d\mathit{\alpha }}$ for parameters $p\in (0,1]$, $\mathit{\alpha }>1$, and $\mathit{\beta }>0$. We show that when $\mathit{\alpha }>1+1∕d$, 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 $exp(-\mathrm{\Theta }(k^{(d-1)∕d}))$. This corresponds to classical results for nearest-neighbor Bernoulli percolation on ${\mathbb{Z}}^d$, but is in contrast to long-range percolation with $\mathit{\alpha }<1+1∕d$, when the exponent of the stretched exponential decay changes to $2-\mathit{\alpha }$. 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.