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.

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with powerlaw exponent τ ϵ (2, 3), sample two vertices ut , vt uniformly at random when the graph has t vertices and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in t ' ≥ t , called the distance evolution. We show that there is a tight strip around the function 4 log log(t)-log(log(t'/t)ν1) | log(τ-2)| ν 2 that the distance evolution never leaves with high probability as t tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a nonnegative random variable L.