Noise sensitivity and Voronoi percolation

TL;DR

This paper investigates noise sensitivity and threshold phenomena in Poisson Voronoi percolation on the plane, demonstrating that crossing events are noise sensitive with a polynomial threshold window using algorithmic and discretization techniques.

Contribution

It introduces a novel approach combining randomized algorithms and discretization to analyze noise sensitivity and thresholds in continuum percolation models.

Findings

Box-crossing events are noise sensitive.

Threshold phenomena occur with polynomial window.

Techniques apply broadly to various models.

Abstract

In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R}^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.