Capacity of loop-erased random walk

2024/11/20 by Maarten Markering, Markering, Maarten
Biochemistry, Genetics and Molecular Biology · #37A25 #60F15 #60G50 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2411.13505

openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the capacity of loop-erased random walk (LERW) on ℤ^d. For d≥4, 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/β, with β 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.

Related

• Four dimensional loop-erased random walk
• Branching capacity of a random walk range
• Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk
• The Hölder continuity of the scaling limit of three-dimensional loop-erased random walk
• The scaling limit of loop-erased random walk in three dimensions