Power rate of convergence of discrete curves: framework and applications

Abstract

We develop a framework for power-law rates of convergence of random discrete curves to Schramm Loewner Evolution (SLE). We show that a power-law rate of convergence of a suitable martingale observable gives a power-law rate of convergence of the interface to SLE, provided the discrete curves satisfy the Kemppainen-Smirnov crossing condition; the proof uses known estimates on the derivative of the SLE map. We apply the framework to the exploration process of critical site percolation on the triangular lattice, obtaining a power-law rate of convergence to SLE${}_6$.

arXiv commentThe exposition was corrected and expanded