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The Fourier spectrum of critical percolation

Christophe Garban CNRS Département de mathématiques (UMPA) École normale supérieure de Lyon, Lyon Cedex 07, France Gábor Pete Department of Mathematics, University of Toronto Oded Schramm Theory Group of Microsoft Research, One Microsoft Way

TBD mathscidoc:1701.332018

Acta Mathematica, 205, (1), 19-104, 2008.5
Consider the indicator function$f$of a 2-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 ${\frac{31}{36}}$ almost surely, and the corresponding dimension in the half-plane is ${\frac{5}{9}}$ . It is also proved that critical bond percolation on the square grid has exceptional times almost surely. 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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@inproceedings{christophe2008the,
  title={The Fourier spectrum of critical percolation},
  author={Christophe Garban, Gábor Pete, and Oded Schramm},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203355295905727},
  booktitle={Acta Mathematica},
  volume={205},
  number={1},
  pages={19-104},
  year={2008},
}
Christophe Garban, Gábor Pete, and Oded Schramm. The Fourier spectrum of critical percolation. 2008. Vol. 205. In Acta Mathematica. pp.19-104. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203355295905727.
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