The Rate Estimates Of Convergence Of Harmonic Explorer Process To SLE₄

The Schramm-Loewner Evolution(SLE) introduced by Schramm in2000is a one-parameter family of random fractal curves, which can be obtained by solving Loewner’sdifferential equation with driving parameter being a one-dimensional Brownian Motion.It is known that the harmonic explorer on vertices of the planar honeycomb latticeconverges to chordal SLE4as the grid gets finer. In this paper the rate estimates ofconvergence of harmonic explorer process to SLE4is discussed. Firstly, we give thedecay of moment estimates for increments of Loewner driving function associated withthe harmonic explorer. Secondly, based on this result and the properties of SLE4, wederive the rate estimates of convergence of the driving function of harmonic explorer toBrownian Motion4. Lastly, we discuss the probabilistic estimates of correspondingconvergence rate in the sense of locally Hausdorff metric. This provides an approach foroptimizing critical exponents of discrete models.