Crossing probabilities for Voronoi percolation

TL;DR

This paper proves that crossing probabilities in Voronoi percolation follow Russo-Seymour-Welsh theory, leading to bounds on crossing probabilities and implications like polynomial decay of the one-arm event at criticality.

Contribution

It establishes the validity of Russo-Seymour-Welsh theory for Voronoi percolation, a significant extension of percolation theory.

Findings

Crossing probabilities are bounded by aspect ratio-dependent constants.

Polynomial decay of the one-arm event at criticality.

Russo-Seymour-Welsh theory applies to Voronoi percolation.

Abstract

We prove that the standard Russo-Seymour-Welsh theory is valid for Voronoi percolation. This implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio. This result has many consequences, such as the polynomial decay of the one-arm event at criticality.