Rate of Convergence for Cardy's Formula

2012/10/15 by Dana Mendelson, Mendelson, Dana, Asaf Nachmias +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR

paper · pdf · doi:10.48550/arxiv.1210.4201

31 pages, 17 figures. Referee corrections made, to appear in Communications in Mathematical Physics

openalex publication_date 2012/10/15 · arxiv created 2014/05/02 · arxiv updated 2014/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)).

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