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 potentially infinite-range weighted random connection models. The vertex set is given by a marked Poisson point process on ${\mathbb{R}}^d$, $d\ge 2$, with intensity $\lambda >0$, where each vertex carries an independent weight. Given a pair of vertices, an edge is drawn independently with a probability that depends on both their spatial displacement and their respective weights. Our geometric and moment assumptions allow the expected degree to be unbounded over the weights. We prove exponential tails for the number of vertices in the root cluster throughout the subcritical regime. We also obtain a linear lower bound on the percolation probability near criticality from above. The class of systems considered includes min-reach models, Poisson--Boolean overlap models, and suitable softened versions thereof. Our proof combines a finite-volume lattice approximation with Hutchcroft's exploration approach to the OSSS inequality.

arXiv comment49 pages, 2 figures. Results generalized; substantial revisions and corrections to the proofs