Gilbert's disc model with geostatistical marking

TL;DR

This work studies Gilbert's disc model in the plane with radii driven by a stationary ergodic field $\varphi$ independent of a Poisson point process, exploring how geostatistical marking affects coverage and percolation compared to the iid-radius case with the same marginal distribution. It develops a probabilistic framework using a zero-one law and novel correlation measures $\pi_\lambda(n)$ and $\overline{\pi}(n)$, along with a finite-size criterion to characterize phase transitions. The authors show that complete coverage under geostatistical marking implies complete coverage in the iid setting, but the spatial dependence can either enhance or suppress long-range connectivity; they demonstrate these effects via concrete cylinder and Voronoi-based constructions. A key contribution is a Voronoi-cell based two-radii construction showing that the critical density $\lambda_c(\mu,p)$ can be below, equal to, or above the iid threshold $\lambda_\Phi(p)$, with precise asymptotics as $\mu\to\infty$ or $\mu\to0$, revealing rich dependencies on the correlation structure of the marking field.

Abstract

We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in $\mathbb{R}^2$ with radii determined by an underlying stationary and ergodic random field $\varphi:\mathbb{R}^2\to[0,\infty)$, independent of the Poisson process. When the random field is independent of the point process one often talks about 'geostatistical marking'. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of $\mathbb{R}^2$ does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

Theorems & Definitions (25)

• Example 1
• Example 2
• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof