Subcritical annulus crossing in spatial random graphs

TL;DR

This paper investigates conditions under which annuli are crossed or not in general continuum percolation models, relating crossing probabilities to long edge occurrences, with applications to weight-dependent random connection models.

Contribution

It establishes a sharp condition for positive critical annulus-crossing intensity in non-monotone, dependent models, extending percolation theory to broader settings.

Findings

Annuli are either not crossed at low intensities or crossed by a single edge.

Derived conditions for long edges based on decay coefficients.

Applied results to weight-dependent random connection models.

Abstract

We consider general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation. In particular, this includes models constructed on general point sets other than the standard Poisson point process or the Bernoulli-percolated lattice. Moreover, in our setting the existence of an edge may depend not only on the two end vertices but also on a surrounding vertex set and models are included that are not monotone in some of their parameters. We study the critical annulus-crossing intensity $\widehat{\lambda}_{c}$, which is smaller or equal to the classical critical percolation intensity $\lambda_{c}$ and derive a condition for $\widehat{\lambda}_{c}>0$ by relating the crossing of annuli to the occurrence of long edges. This condition is sharp for models that have a modicum of independence. In a nutshell, our result states that annuli are either not crossed for small intensities or crossed by a single edge. Our proof rests on a multiscale argument that further allows us to directly describe the decay of the annulus-crossing probability with the decay of long edges probabilities. We apply our result to a number of examples from the literature. Most importantly, we extensively discuss the weight-dependent random connection model in a generalised version, for which we derive sufficient conditions for the presence or absence of long edges that are typically easy to check. These conditions are built on a decay coefficient $\zeta$ that has recently seen some attention due to its importance for various proofs of global graph properties.