Capacity of loop-erased random walk

TL;DR

This paper investigates the capacity of loop-erased random walk (LERW) in various dimensions, establishing laws, limits, and scaling behaviors, and connecting these to known models and exponents.

Contribution

It provides explicit limit expressions for LERW capacity in high dimensions and characterizes the scaling limit in three dimensions, including ergodicity and randomness results.

Findings

Strong law of large numbers for d≥4

Explicit limit expressions involving non-intersection probabilities

Scaling limit of capacity in 3D is random

Abstract

We study the capacity of loop-erased random walk (LERW) on $\mathbb{Z}^d$. For $d\geq4$, we prove a strong law of large numbers and give explicit expressions for the limit in terms of the non-intersection probabilities of a simple random walk and a two-sided LERW. Along the way, we show that four-dimensional LERW is ergodic. For $d=3$, we show that the scaling limit of the capacity of LERW is random. We show that the capacity of the first $n$ steps of LERW is of order $n^{1/\beta}$, with $\beta$ the growth exponent of three-dimensional LERW. We express the scaling limit of the capacity of LERW in terms of the capacity of Kozma's scaling limit of LERW. As a corollary, we obtain the scaling limit of the LERW in three dimensions when parametrized by its capacity.