Cluster-size decay in supercritical long-range percolation

arXiv2303.00712MaRDI QIDQ6509009

Júlia Komjáthy, Joost Jorritsma, Dieter Mitsche

Abstract: We study the cluster-size distribution of supercritical long-range percolation on $mathbb{Z}^d$, where two vertices $x,yinmathbb{Z}^d$ are connected by an edge with probability $$ for parameters $pin(0,1)$, $alpha>1$, and $$. We show that when $alpha>1+1/d$, and either $$ or $p$ is sufficiently large, the probability that the origin is in a finite cluster of size at least $k$ decays as $$. This corresponds to classical results for nearest-neighbor Bernoulli percolation on $mathbb{Z}^d$, but is in contrast to long-range percolation with $alpha<1+1/d$, when the exponent of the stretched exponential decay changes to $2-alpha$. 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.

Mathematics Subject Classification ID

Interacting random processes; statistical mechanics type models; percolation theory (60K35) Combinatorial probability (60C05)

This page was built for publication: Cluster-size decay in supercritical long-range percolation

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6509009)