Widths of crossings in Poisson Boolean percolation

TL;DR

This paper investigates the width of crossings in planar Poisson Boolean percolation, revealing how it varies across different regimes and between occupied and vacant sets, providing insights into the geometric properties of these percolation models.

Contribution

It provides a detailed analysis of crossing widths in Poisson Boolean percolation, distinguishing behaviors across critical, sub-critical, and super-critical regimes for both occupied and vacant sets.

Findings

Crossing widths differ significantly across regimes.

Occupied and vacant sets exhibit distinct crossing width behaviors.

Results enhance understanding of geometric properties in percolation models.

Abstract

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model does contain a crossing of an $n\times n$ square, how wide is this crossing? The answer depends on the whether we consider the critical, sub- or super-critical regime, and is different for the occupied and vacant sets.