Acta Mathematica

The Fourier spectrum of critical percolation

Christophe Garban, Gábor Pete, and Oded Schramm

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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.

Note

CG was partially supported by the ANR under the grant ANR-06-BLAN-0058. GP was partially supported by the Hungarian National Foundation for Scientific Research, grant T049398, and by an NSERC Discovery Grant. For large parts of the work, all three authors were at Microsoft Research.

Note

Deceased on September 1st, 2008 (Oded Schramm).

Article information

Source
Acta Math., Volume 205, Number 1 (2010), 19-104.

Dates
Received: 6 May 2008
Revised: 12 November 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892483

Digital Object Identifier
doi:10.1007/s11511-010-0051-x

Mathematical Reviews number (MathSciNet)
MR2736153

Zentralblatt MATH identifier
1219.60084

Rights
2010 © Institut Mittag-Leffler

Citation

Garban, Christophe; Pete, Gábor; Schramm, Oded. The Fourier spectrum of critical percolation. Acta Math. 205 (2010), no. 1, 19--104. doi:10.1007/s11511-010-0051-x. https://projecteuclid.org/euclid.acta/1485892483


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