Scale-free percolation

We formulate and study a model for inhomogeneous long-range percolation on Z(d). Each vertex x is an element of Z(d) is assigned a non-negative weight W-x, where (W-x)(x is an element of Zd) are i.i.d. random variables. Conditionally on the weights, and given two parameters alpha, lambda > 0, the edges are independent and the probability that there is an edge between x and y is given by p(xy) = 1 - exp{-lambda WxWY/vertical bar x - y vertical bar(alpha)}. The parameter lambda is the percolation parameter, while a describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of W-x is regularly varying with exponent tau - 1, then the tail of the degree distribution is regularly varying with exponent gamma = alpha (tau - 1)/d. The parameter gamma turns out to be crucial for the behavior of the model. Conditions on the weight distribution and gamma are formulated for the existence of a critical vale lambda(c) is an element of (0, infinity) such that the graph contains an infinite component when lambda > lambda(c) and no infinite component when lambda > lambda(c). Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point gamma = 2, that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.