Sharpness of the percolation phase transition for weighted random connection models

Abstract

We establish the sharpness of the percolation phase transition for a class of infinite-range weighted random connection models. The vertex set is given by a marked Poisson point process on $\mathbb{R}^d$ with intensity $λ>0$, where each vertex carries an independent weight. Pairs of vertices are then connected independently with a probability that depends on both their spatial displacement and their respective weights. It is well known that such models undergo a phase transition in $λ$ with respect to the existence of an infinite cluster (under suitable assumptions on the connection probabilities and the weight distribution). We prove that in the subcritical regime the cluster-size distribution has exponentially decaying tails, whereas in the supercritical regime the percolation probability grows at least linearly with respect to $λ$ near criticality. Our proof follows the approach of Duminil-Copin, Raoufi, and Tassion, applying the OSSS inequality to a finite-lattice approximation of the continuum model in order to derive a new differential inequality, which we then analyze and pass to the limit. In addition to the classical random connection model, we consider weighted models with unbounded weights satisfying the min-reach condition under which the neighborhood of each vertex is deterministically bounded by a radius depending solely on its weight. Notably, finite range is not assumed -- that is, we allow unbounded edge lengths -- but the weight distribution is required to satisfy appropriate moment conditions. We expect that our method extends to a broad class of weighted random connection models.