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The Fourier spectrum of critical percolation. (English) Zbl 1219.60084
The authors study the harmonic analysis of functions arising from planar percolation and answer all of the previously posted problems regarding their Fourier expansions. Namely, consider the indicator function \(f\) of a two-dimensional percolation event. The Fourier transform of the function is studied and sharp bounds are obtained for its lower tail in several situations. The authors also derive some applications to the behavior of percolation under noise and to the study of dynamic percolation. They show that the set of exceptional times of dynamical critical site percolation on a triangular grid in which the origin percolates has dimension 31/36 almost surely, and the corresponding dimension in the half-plane is 5/9. In addition, the asymptotic 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. The technique introduced in this paper seems also to be helpful in the harmonic analysis of other functions.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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