Quantitative noise sensitivity and exceptional times for percolation

• 成果类型: Article
• 署名作者: Schramm, Oded; Steif, Jeffrey E.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.171.619
• 发表日期: 2010
• 页码: 619-672
• 关键词: critical exponents; MARKOV-PROCESSES; SCALING LIMITS

摘要

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.