We study cluster sizes in supercritical d-dimensional inhomogeneous percolation models with long-range edges —such as long-range percolation— and/or heavy-tailed degree distributions —such as geometric inhomogeneous random graphs and the age-dependent random connection model. Our focus is on large deviations of the size of the largest cluster in the graph restricted to a finite box as its volume tends to infinity. Compared to nearest neighbor Bernoulli bond percolation on Zd, we show that long edges can increase the exponent of the polynomial speed of the lower tail from (d-1)/d to any ζ⋆∈((d-1)/d,1). We prove that this exponent ζ⋆ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin C(0). For the upper tail of large deviations, we prove that its speed is logarithmic for models with power-law degree distributions. We express the rate function via the generating function of |C(0)|. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.

We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent ζ ∈ (0, 1) of the decay of the cluster-size distribution. That is, with (Formula presented) denoting the number of vertices in the component of the vertex at (Formula presented), we prove (Formula presented) The value of ζ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension d, the power-law tail exponent τ of the degree distribution and a long-range parameter α governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: ζ in this regime depends both on τ, α. We also prove that the second-largest component in a box of volume n is of size (Formula presented) with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space.

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.