Acta Mathematica

, Volume 205, Issue 1, pp 19–104

The Fourier spectrum of critical percolation

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.

Copyright information

© Institut Mittag-Leffler 2010

Authors and Affiliations

  1. 1.CNRS Département de mathématiques (UMPA) École normale supérieure de LyonLyon Cedex 07France
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.Theory Group of Microsoft ResearchOne Microsoft WayRedmondU.S.A.