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One goal of this paper is to prove that dynamical critical site percolation on the planar triangular lattice has exceptional times at which percolation occurs. In doing so, new quantitative noise sensitivity results for percolation are obtained. The latter is based on a novel method for controlling the “level $k$” Fourier coefficients via the construction of a randomized algorithm which looks at random bits, outputs the value of a particular function but looks at any fixed input bit with low probability. We also obtain upper and lower bounds on the Hausdorff dimension of the set of percolating times. We then study the problem of exceptional times for certain “$k$-arm” events on wedges and cones. As a corollary of this analysis, we prove, among other things, that there are no times at which there are two infinite “white” clusters, obtain an upper bound on the Hausdorff dimension of the set of times at which there are both an infinite white cluster and an infinite black cluster and prove that for dynamical critical bond percolation on the square grid there are no exceptional times at which three disjoint infinite clusters are present.
@article{10_4007_annals_2010_171_619,
author = {Oded Schramm and Jeffrey E. Steif},
title = {Quantitative noise sensitivity and exceptional times for percolation},
journal = {Annals of mathematics},
pages = {619--672},
publisher = {mathdoc},
volume = {171},
number = {2},
year = {2010},
doi = {10.4007/annals.2010.171.619},
mrnumber = {2630053},
zbl = {1213.60160},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.619/}
}
TY - JOUR AU - Oded Schramm AU - Jeffrey E. Steif TI - Quantitative noise sensitivity and exceptional times for percolation JO - Annals of mathematics PY - 2010 SP - 619 EP - 672 VL - 171 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.619/ DO - 10.4007/annals.2010.171.619 LA - en ID - 10_4007_annals_2010_171_619 ER -
%0 Journal Article %A Oded Schramm %A Jeffrey E. Steif %T Quantitative noise sensitivity and exceptional times for percolation %J Annals of mathematics %D 2010 %P 619-672 %V 171 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.619/ %R 10.4007/annals.2010.171.619 %G en %F 10_4007_annals_2010_171_619
Oded Schramm; Jeffrey E. Steif. Quantitative noise sensitivity and exceptional times for percolation. Annals of mathematics, Tome 171 (2010) no. 2, pp. 619-672. doi: 10.4007/annals.2010.171.619
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