One-point function estimates for loop-erased random walk in three dimensions

TL;DR

This paper provides asymptotic estimates for the one-point function and non-intersection probabilities of loop-erased random walk in three dimensions, aiding the understanding of its scaling limit.

Contribution

It introduces new asymptotic estimates for LERW in three dimensions and develops a coupling technique for LERW and SRW with different starting points.

Findings

Asymptotic estimates for one-point functions in 3D LERW

Non-intersection probability bounds for LERW and SRW

Coupling method for LERW and SRW with different initial conditions

Abstract

In this work, we consider loop-erased random walk (LERW) in three dimensions and give an asymptotic estimate on the one-point function for LERW and the non-intersection probability of LERW and simple random walk in three dimensions for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.