Noise Sensitivity in Continuum Percolation

TL;DR

This paper proves that the Poisson Boolean model in continuum percolation is noise sensitive at criticality, marking the first such result for this model and for cases where the critical probability differs from 1/2.

Contribution

It introduces the first noise sensitivity proof for continuum percolation models and develops a general approximation method using discrete models with bounded critical probability.

Findings

Poisson Boolean model is noise sensitive at criticality

Develops a method to approximate continuum models by discrete models

Extends the Benjamini-Kalai-Schramm Theorem to biased measures

Abstract

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p_c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.