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 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.

68 pages, 13 figures. To appear in Probab. Theory Relat. Fields

References in corpus (2)

• Convergence of radial loop-erased random walk in the natural parametrization
• One-point function estimates for loop-erased random walk in three dimensions

Cited by in corpus (1)

• Scaling limits of the three-dimensional uniform spanning tree and associated random walk