# Gilbert's disc model with geostatistical marking

## TL;DR

This paper investigates a variant of Gilbert's disc model where disc radii are determined by a stationary ergodic random field, analyzing how spatial dependence affects coverage and percolation properties compared to independent radii models.

## Contribution

It introduces and analyzes a geostatistical marking approach in Gilbert's model, highlighting how spatial dependence influences coverage and percolation thresholds.

## Key findings

- Complete coverage may not occur simultaneously across the plane.
- Spatial dependence can both raise and lower percolation thresholds.
- Dependence affects the occurrence of long-range connections.

## 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.