We study the cluster-size distribution of supercritical long-range percolation on , where two vertices are connected by an edge with probability for parameters , , and . We show that when , 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 . This corresponds to classical results for nearest-neighbor Bernoulli percolation on , but is in contrast to long-range percolation with , when the exponent of the stretched exponential decay changes to . 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.
Electron. J. Probab.
29:
1-36
(2024).
DOI: 10.1214/24-EJP1135