Noise sensitivity of percolation via differential inequalities

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