Sharpness of the percolation phase transition for weighted random connection models

TL;DR

This paper proves the sharpness of the phase transition in weighted random connection models with infinite-range edges, showing exponential decay of cluster sizes below criticality and linear growth of percolation probability above.

Contribution

It introduces a novel approach using the OSSS inequality to establish sharp phase transition properties for a broad class of weighted continuum percolation models.

Findings

Exponential decay of cluster sizes in the subcritical regime.

Percolation probability grows at least linearly near criticality.

Method extends to models with unbounded weights and edges.

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 $\lambda>0$, 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 $\lambda$ 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 $\lambda$ 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.