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

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 improves the previous work of Gady Kozma (Acta Math. 199(1):29-152), which shows that the rescaled trace of 3D LERW converges weakly to a random compact set with respect to the Hausdorff distance.Comment: 74 pages, 3 figure