Acta Mathematica

, Volume 205, Issue 1, pp 19–104

The Fourier spectrum of critical percolation

Authors

  • Christophe Garban
    • CNRS Département de mathématiques (UMPA) École normale supérieure de Lyon
    • Department of MathematicsUniversity of Toronto
  • Oded Schramm
    • Theory Group of Microsoft ResearchOne Microsoft Way
Article

DOI: 10.1007/s11511-010-0051-x

Cite this article as:
Garban, C., Pete, G. & Schramm, O. Acta Math (2010) 205: 19. doi:10.1007/s11511-010-0051-x

Abstract

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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© Institut Mittag-Leffler 2010