Quantitative noise sensitivity and exceptional times for percolation

TL;DR

This paper proves the existence of exceptional times in dynamical critical site percolation on the triangular lattice, introduces new noise sensitivity techniques, and analyzes the Hausdorff dimension of these times for various percolation events.

Contribution

It introduces a novel method for controlling Fourier coefficients via randomized algorithms and establishes quantitative noise sensitivity results for percolation.

Findings

Existence of exceptional times where percolation occurs on the triangular lattice.

Bounds on the Hausdorff dimension of the set of percolating times.

No exceptional times with multiple infinite clusters in certain configurations.

Abstract

One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the "level k" Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain "k-arm" events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite "white" clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infinite clusters are present.