Power rate of convergence of discrete curves: framework and applications

TL;DR

This paper develops a general framework to estimate the convergence rates of discrete random curves to SLE curves, applying it to critical percolation on hexagonal lattices, and establishing power-law convergence rates.

Contribution

It introduces a unified approach linking crossing event bounds and martingale observables to convergence rates of discrete curves to SLE.

Findings

Power-law convergence rate for interface to SLE curves.

Application to critical site percolation on hexagonal lattice.

Convergence of exploration process to SLE6 with explicit rate.

Abstract

We provide a general framework of estimates for convergence rates of random discrete model curves approaching Schramm Loewner Evolution (SLE) curves in the lattice size scaling limit. We show that a power-law convergence rate of an interface to an SLE curve can be derived from a power-law convergence rate for an appropriate martingale observable provided the discrete curve satisfies a specific bound on crossing events, the Kempannien-Smirnov condition, along with an estimate on the growth of the derivative of the SLE curve. We apply our framework to show that the exploration process for critical site percolation on hexagonal lattice converges to the SLE$_6$ curve with a power-law convergence rate.