Some partial results on the convergence of loop-erased random walk to SLE(2) in the natural parametrization

TL;DR

This paper proposes a strategy to prove the convergence of loop-erased random walk on Z^2 to SLE(2) in the natural parametrization, extending previous methods and establishing partial results including a new estimate.

Contribution

It introduces a new approach to demonstrate convergence in the natural parametrization, emphasizing occupation measure convergence and providing partial results and estimates.

Findings

Established partial results towards occupation measure convergence.

Provided a new estimate for loop-erased random walk behavior.

Outlined a strategy extending previous convergence proofs to natural parametrization.

Abstract

We outline a strategy for showing convergence of loop-erased random walk on the Z^2 square lattice to SLE(2), in the supremum norm topology that takes the time parametrization of the curves into account. The discrete curves are parametrized so that the walker moves at a constant speed determined by the lattice spacing, and the SLE(2) curve has the recently introduced natural time parametrization. Our strategy can be seen as an extension of the one used by Lawler, Schramm, and Werner to prove convergence modulo time parametrization. The crucial extra step is showing that the expected occupation measure of the discrete curve, properly renormalized by the chosen time parametrization, converges to the occupation density of the SLE(2) curve, the so-called SLE Green's function. Although we do not prove this convergence, we rigorously establish some partial results in this direction including a new loop-erased random walk estimate.