Power rate of convergence of discrete curves: framework and applications

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.