The H\"older continuity of the scaling limit of three-dimensional loop-erased random walk

TL;DR

This paper proves that the scaling limit of three-dimensional loop-erased random walk exhibits a specific H"older continuity property, being almost surely $h$-H"older continuous for all $h < 1/eta$ and not for $h = 1/eta$, revealing detailed regularity characteristics.

Contribution

It establishes the precise H"older continuity exponents of the 3D LERW scaling limit, advancing understanding of its regularity properties.

Findings

The scaling limit is $h$-H"older continuous for all $h < 1/eta$.

The scaling limit is not $1/eta$-H"older continuous.

Provides a rigorous characterization of the regularity of 3D LERW scaling limit.

Abstract

Let $\beta$ be the growth exponent of the loop-erased random walk (LERW) in three dimensions. We prove that the scaling limit of 3D LERW is $h$-H\"older continuous almost surely for all $h < 1/\beta$, while not $1/\beta$-H\"older continuous almost surely.