Quantitative noise sensitivity and exceptional times for percolation

Abstract

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 3 disjoint infinite clusters are present.

The Fourier spectrum of critical percolation

“…This was proved in [K2]; see [SSt,Proposition A.1] and [N, §4] for concise proofs. The above properties in particular give for r<r <R <R that…”

“…(This can be proved by combining the above arguments with the proof of [SSt,Proposition A.5]. ) Proposition 5.9.…”

“…The main reason for this is a classical arm separation phenomenon, see e.g. [SSt,Appendix,Lemma A.2], which roughly says that conditioned on having four arms connecting ∂B to the appropriate boundary arcs on ∂Q, with positive conditional probability, these arms are "well separated" on ∂B. On this event, a positive proportion of the ω B configurations enable the crossing of Q, while a positive proportion disable all crossings.…”

“…Let Z denote the event that in ω for each k∈{0, 2, 4, 6} there is a crossing from ∂B(0, r 0 ) to ∂B(0, 8r) in L k ∪A(r 0 , 4r), which is white when k∈{0, 4} and black when k∈{2, 6}. By the proof of [SSt,Lemma A.3], we know (see Figure 5.1) that there is a constant c 0 =c 0 (M )>0 such that…”

“…for some fixed constants C, ε>0 and every 1 r R. The left inequality can be obtained by combining α 5 (r, R) (r/R) 2 (see [KSZ,Lemma 5] or [SSt,Corollary A.8]), the RSW estimate α 1 (r, R)<O(1)(r/R) ε , and, finally, the relation α 1 (r, R)α 4 (r, R) α 5 (r, R) (which follows from Reimer's inequality [R], or from Proposition A.1 in our appendix). The right-hand inequality in (2.6) follows from [K2]; see also [BKS,Remark 4.2].…”