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Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk

Sarai Hernandez-Torres, Xinyi Li, Daisuke Shiraishi

TL;DR

This work establishes that the scaling limit of three-dimensional loop-erased random walk (LERW) carries a natural, intrinsic parametrization given by the β-dimensional Minkowski content of the limit trace, with β in (1,5/3]. It proves the existence of a limiting occupation measure and shows it is proportional to the Minkowski content, yielding a canonical parametrization of the limit curve that is invariant under scaling and rotations. The authors develop sharp one-point and ball-hitting asymptotics across scales, prove the existence and continuum form of a two-point function, and use a continuity property of ball hitting to connect discrete LERW to its continuum limit. Together, these results deepen the understanding of the 3D LERW scaling limit and its fractal-geometric structure, and they establish a robust framework for a natural parametrization via Minkowski content.

Abstract

In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its $β$-dimensional Minkowski content, where $β\in (1, 5/3]$ is its Hausdorff dimension. In doing this we also establish the existence of the two-point function and provide some sharp estimates on one-point function and ball-hitting probabilities for 3D LERW in any scale, which is a considerable improvement of previous results.

Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk

TL;DR

This work establishes that the scaling limit of three-dimensional loop-erased random walk (LERW) carries a natural, intrinsic parametrization given by the β-dimensional Minkowski content of the limit trace, with β in (1,5/3]. It proves the existence of a limiting occupation measure and shows it is proportional to the Minkowski content, yielding a canonical parametrization of the limit curve that is invariant under scaling and rotations. The authors develop sharp one-point and ball-hitting asymptotics across scales, prove the existence and continuum form of a two-point function, and use a continuity property of ball hitting to connect discrete LERW to its continuum limit. Together, these results deepen the understanding of the 3D LERW scaling limit and its fractal-geometric structure, and they establish a robust framework for a natural parametrization via Minkowski content.

Abstract

In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its -dimensional Minkowski content, where is its Hausdorff dimension. In doing this we also establish the existence of the two-point function and provide some sharp estimates on one-point function and ball-hitting probabilities for 3D LERW in any scale, which is a considerable improvement of previous results.
Paper Structure (25 sections, 37 theorems, 223 equations, 2 figures)

This paper contains 25 sections, 37 theorems, 223 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof overview
  4. Organization of this work
  5. Notation and some useful facts
  6. General notation
  7. Metric spaces
  8. Minkowski content
  9. Weak convergence of probability measures
  10. The scaling limit of 3D LERW
  11. Up-to-constant one- and two-point estimates
  12. Asymptotic independence for 3D LERW
  13. A continuity result
  14. Sharp one-point function estimates
  15. Proof of Theorem \ref{['thm:oneptnew']}
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exists a universal constant $c_0>0$ such that for any dyadic box $V \in {\cal D}^o$, ${\rm Cont}_\beta({\cal K} \cap V)$ exists and equals to $c_0\mu(V)$ almost surely.

Figures (2)

  • Figure 1: The equator $\ell$ divides $\partial B$ into two shells: the northern hemisphere $F_{+}^3$ and the southern one $F = F_{-}^3$. The shaded area represents the set $F' \subset \partial F$, consisting of points $x \in F$ in the southern hemisphere that are at a distance greater than $\zeta = \varepsilon^{b/20}$ from the equatorial line $\ell$.
  • Figure 2: An illustration of the event $\{ u_1 \neq u_1' \}$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 48 more