Quantitative noise sensitivity and exceptional times for percolation

By Oded Schramm, Jeffrey E. Steif. 2005

148 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0b656831940b3491e77efb73871742b0410eac5c

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Citations in the corpus

Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “Though there is some partial understanding of the Fourier spectrum of percolation [82], the complete picture is unclear.”
Sharp phase transition for the random-cluster and Potts models via decision trees (2017, methodology): “In particular, random decision trees (sometimes called randomized algorithms) were used in [SS10] to study the noise sensitivity of Boolean functions, for instance in the context of percolation theory.”
Near-critical percolation in two dimensions (2007, methodology): “Let us also mention that the techniques presented here are important to study various models related to the critical regime, for instance Incipient Infinite Clusters[17; 24; 25], Dynamical Percolation [ 45 ], Gradient Percolation[39]...”
Lectures on two-dimensional critical percolation (2007, background): “These questions on near-critical percolation are related to how percolation events depend on ﬂuctuations or noising of the discrete realization. For results and references in this direction, see e.g. [39]. • It is still an open problem to generalize these ideas (Kesten’s scaling relations) to depend”