On the rate of convergence of loop-erased random walk to SLE(2)

TL;DR

This paper establishes a quantitative rate at which loop-erased random walk converges to SLE(2), improving understanding of their relationship and providing explicit convergence metrics.

Contribution

It introduces a new estimate for the difference between discrete and continuous Green's functions, leading to a quantifiable convergence rate for LERW to SLE(2).

Findings

Derived a convergence rate for the Loewner driving function to Brownian motion.

Established a convergence rate for the paths in Hausdorff distance.

Improved estimates over existing results for specific domain classes.

Abstract

We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE(2), we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.