Noise sensitivity of percolation via differential inequalities

TL;DR

This paper presents a new proof of the sharp noise sensitivity of critical percolation without spectral tools, using differential inequalities, and also establishes the Hausdorff dimension of certain dynamical percolation times.

Contribution

It introduces a novel proof technique for noise sensitivity in percolation based on differential inequalities, avoiding spectral methods, and provides new results on the Hausdorff dimension of dynamical percolation times.

Findings

New proof of noise sensitivity without spectral tools

Hausdorff dimension of times with primal and dual percolation equals 2/3

Differential inequalities approach inspired by Kesten's scaling relations

Abstract

Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete and Schramm. Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four-arm event, in the spirit of Kesten's proof of scaling relations. We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals $2/3$ a.s.