2010/03/25 by Oded Schramm, Jeffrey E. Steif · 3 citations
Mathematics · #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals #Markov Chains and Monte Carlo Methods #Mathematics #Hausdorff dimension #Disjoint sets #Combinatorics #White noise #Upper and lower bounds #Percolation critical exponents #Square tiling #Percolation (cognitive psychology) #Discrete mathematics #Mathematical analysis #Critical exponent #Grid #Geometry #Statistics #Scaling
paper · pdf · doi:10.4007/annals.2010.171.619
openalex publication_date 2010/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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 "karm" 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. 620 ODED SCHRAMM and JEFFREY E. STEIF 6. Hausdorff dimension of exceptional times 7. Exceptional times for k-arm events 8. Upper bounds for k-arm times 9. The square lattice 10. Some open questions