Convergence of three-dimensional loop-erased random walk in the natural parametrization

TL;DR

This paper proves that the three-dimensional loop-erased random walk, when scaled and parametrized by length, converges to its continuous scaling limit, enhancing previous weak convergence results.

Contribution

It establishes the convergence of 3D LERW in the natural parametrization with respect to the uniform topology, improving prior weak convergence results.

Findings

3D LERW converges to its scaling limit in natural parametrization.

Convergence is with respect to the uniform topology.

Results strengthen previous weak convergence findings.

Abstract

In this work we consider loop-erased random walk (LERW) and its scaling limit in three dimensions, and prove that 3D LERW parametrized by renormalized length converges to its scaling limit parametrized by some suitable measure with respect to the uniform convergence topology in the lattice size scaling limit. Our result greatly improves the work (Acta Math. 199(1):29-152) of Gady Kozma which establishes the weak convergence of the rescaled trace of 3D LERW towards a random compact set with respect to the Hausdorff distance.