Crossing probabilities in geometric inhomogeneous random graphs

Abstract

In a geometric inhomogeneous random graph vertices are given by the points of a Poisson process and are equipped with independent weights following a heavy tailed distribution. Any pair of distinct vertices is independently forming an edge with a probability decaying as a function of the product of the weights divided by the distance of the vertices. For this continuum percolation model we study the crossing probabilities of annuli, i.e. the probabilities that there exist paths starting inside a ball and ending outside a larger concentric ball with increasing inner and outer radii. Depending on the radii, the power-law exponent of the degree distribution and the decay of the probability of long edges, we identify regimes where the crossing probabilities by a path are equivalent to the crossing probabilities by one or by two edges. We also identify the escape probabilities from balls with strong centre, i.e. the asymptotics of the probability that there exists a path starting from a vertex with a given weight leaving a centred ball as radius and weight are going to infinity. As a corollary we get the subcritical one-arm exponents characterising the decay of the probability that a typical point is in a component not contained in a centred ball whose radius goes to infinity.