Noise sensitivity of percolation via differential inequalities

Abstract

Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete, and Schramm (Acta Math. 205 (2010), 19–104). Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four-arm event, in the spirit of Kesten's proof of scaling relations in Kesten (Comm. Math. Phys. 109 (1987), 109–156). We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals 2 / 3 $2/3$ almost surely.