Sharpness of the percolation phase transition for weighted random connection models

TL;DR

This work proves sharpness of the percolation phase transition for a broad class of infinite-range weighted random connection models on marked Poisson point processes. It combines a finite-volume discretization with an OSSS-based exploration, a Russo-type derivative framework, and a careful limiting procedure to derive differential inequalities for cluster tails and susceptibility. The min-reach RCM, which permits unbounded weights and long-range edges under moment conditions, is shown to satisfy the core assumptions, enabling sharpness results that extend beyond classical unweighted or bounded-weight settings. The results yield exponential decay in the subcritical regime and at least linear growth of the percolation probability near criticality, establishing the coincidence of the two critical intensities and providing a robust methodology for a wide range of weighted continuum percolation 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.

Figures (2)

• Figure 1: Realization of a non-weighted random connection model in $\mathbb{R}^2$ with the adjacency function given by $\phi(r) \;=\; 1 - \exp\bigl(- r^{-3}\bigr)$.
• Figure 2: Realization of a min-reach random connection model in $\mathbb R^2$ with a Pareto-tailed weight distribution $\pi$. Displayed node radii are proportional to the weights. The adjacency function is given by $\varphi(r,a,b) = \mathbf{1}_{\{r\le \min(a,b)\}}\, \bigl(1 - e^{-\,ab/r^{3}}\bigr).$

Theorems & Definitions (69)

• Definition 1.1: Cluster of a point
• Remark 1.2
• Definition 1.3: Tail function, percolation probability, expected susceptibility
• Definition 1.4: Critical intensities
• Lemma 1.5: Non-triviality of the phase transition
• Remark 1.6
• Remark 1.7
• Theorem 1.8: Sharpness of the phase transition
• Remark 1.9
• Remark 2.1