The Fourier spectrum of critical percolation

TL;DR

This paper analyzes the Fourier spectrum of a percolation crossing indicator function, deriving bounds and applying them to determine the dimension of exceptional times and the sensitivity of configurations.

Contribution

It provides sharp bounds on the Fourier spectrum of percolation crossing functions and applies these results to analyze exceptional times and configuration perturbations.

Findings

Dimension of exceptional times in dynamical critical site percolation is 31/36.

Dimension in the half-plane is 5/9.

Critical bond percolation on the square grid has exceptional times almost surely.

Abstract

Consider the indicator function $f$ of a two-dimensional percolation crossing event. In this paper, the Fourier transform of $f$ is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the half-plane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.