Continuum Percolation in a Nonstabilizing Environment

TL;DR

This paper investigates phase transitions in continuum percolation within a nonstabilizing environment modeled by a Cox process influenced by a planar Poisson line process, revealing the absence of a sharp phase transition due to long-range dependencies.

Contribution

It establishes phase transition results for a Cox Boolean model with a nonstabilizing environment, extending percolation theory to complex dependent structures.

Findings

No sharp phase transition exists in the model.

Phase transitions are characterized under varying parameters.

Discretization and lattice comparison techniques are employed.

Abstract

We prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in Hoffman 2005.