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 $\mathrm{p}(\Vert x-y\Vert ):=p\min (1,{\hat{I}}^2\Vert x-y\Vert {)}^{-d\hat{I}\pm }$ for parameters $p\in (0,1]$, $\hat{I}\pm >1$, and ${\hat{I}}^2>0$. We show that when $\hat{I}\pm >1+1/d$, and either ${\hat{I}}^2$ or $p$ is sufficiently large, the probability that the origin is in a finite cluster of size at least $k$ decays as $\exp (-\hat{I}˜(k^{(d-1)/d}))$. This corresponds to classical results for nearest-neighbor Bernoulli percolation on ${\mathbb{Z}}^d$, but is in contrast to long-range percolation with $\hat{I}\pm <1+1/d$, when the exponent of the stretched exponential decay changes to $2-\hat{I}\pm$. 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.

36 pages, minor revision: to appear in Electronic Journal of Probability