Convergence in natural parametrization of random walk frontier

TL;DR

The paper proves that the frontier of planar random walk converges to the frontier of planar Brownian motion under natural parametrization, with a concurrent convergence of the renormalized occupation measure. The approach combines deep discrete-to-continuum comparisons via Skorokhod embedding and KMT coupling, detailed moment estimates for frontier points and disks, and a robust Green's-function framework for frontier structures. A two-pronged strategy first establishes modulo time-parametrization convergence (via tightness and Hausdorff convergence) and then upgrades to full natural parametrization by leveraging occupation-measure convergence. The core contributions include sharp one- and two-point frontier Green's function estimates, tight control of frontier-disk versus frontier-point events, and a rigorous pathway to convergence in the natural parametrization, strengthening the link between discrete random walks and Brownian frontier geometry. Collectively, the results illuminate the multifractal nature of the frontier and provide tools for precise discrete-continuum comparisons in two dimensions, with potential applications to related stochastic growth models and SLE-type frontiers.

Abstract

In this paper, we show that the frontier of planar random walk converges weakly under natural parametrization to that of planar Brownian motion. As an intermediate result, we also show the convergence of the renormalized occupation measure.

Figures (4)

• Figure 6.1: Illustration of backtracking and four arms crossing the annulus. Note that typically these arms intersect each other (but we choose not to draw this scenario to avoid a messy figure).
• Figure A.1: Lemma A.1. The radii of two dashed circles are $ce^{15m/16}$ and $ce^{31m/32}$ respectively.
• Figure A.2: Lemma A.2. The radii of two dashed circles are $ce^{15m/16}$ and $ce^{31m/32}$ respectively.
• Figure A.3: Lemma A.3. The radii of two dashed circles are $e^{15m/16}+e^{15m'/16}$ and $e^{m/32}(e^{15m/16}+e^{15m'/16})$ respectively.

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: One-arm non-disconnection estimates
• Lemma 2.2
• Lemma 2.3: Separation lemma, continuous case
• Remark 2.4
• Lemma 2.5: Separation lemma, discrete case
• Remark 2.6
• Lemma 2.7: Skorokhod embedding
• Lemma 2.8: KMT coupling