Rate of Convergence for Cardy’s Formula

We show that crossing probabilities in 2D critical site percolation on the triangular lattice in a piecewise analytic Jordan domain converge with power law rate in the mesh size to their limit given by the Cardy–Smirnov formula. We use this result to obtain new upper and lower bounds of e^O(√( log log R)) R^-1/3 for the probability that the cluster at the origin in the half-plane has diameter R , improving the previously known estimate of R −1/3+ o (1) .