The Fourier spectrum of critical percolation

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.

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