Rate of Convergence for Cardy's Formula

TL;DR

This paper establishes the power law rate of convergence for crossing probabilities in 2D critical site percolation to Cardy's formula and derives improved bounds for cluster diameter probabilities.

Contribution

It provides the first quantitative rate of convergence for Cardy's formula and refines bounds on cluster diameter probabilities in critical percolation.

Findings

Crossing probabilities converge with a power law rate in mesh size.

New bounds for cluster diameter probability are established.

Improved estimates for the probability that a cluster has diameter R.

Abstract

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 exp(O(sqrt(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)).