Convergence of three-dimensional loop-erased random walk in the natural parametrization

Abstract

In this work we consider loop-erased random walk (LERW) and its scaling limit in three dimensions, and prove that 3D LERW parametrized by renormalized length converges to its scaling limit parametrized by some suitable measure with respect to the uniform convergence topology in the lattice size scaling limit. Our result greatly improves the work (Acta Math. 199(1):29-152) of Gady Kozma which establishes the weak convergence of the rescaled trace of 3D LERW towards a random compact set with respect to the Hausdorff distance.

Figures (13)

• Figure 1: Illustration for the hittability of the LERW. The thick solid curve stands for $\lambda^T$ while the thick dotted curve represents $R [ 0, T_{0, r}^{R} ]$ (left) and $R [ 0, T_{v, r}^{R} ]$ (right).
• Figure 2: Setup for the event $F_{y,z}$. The solid simple curve starting at $y$ stands for $\lambda$. The dotted curve is $Y^{2} [0, \tau_{z}]$ which also starts at $y$. The remaining curve started at $z$ represents $Y^{3} [0, T]$.
• Figure 3: Illustration for the events $\widehat{H} \cap (\widehat{H}' )^{c} \cap G^{2}$ and $A$. The dotted curve stands for $\widehat{\gamma}$ and the solid curve represents $R^{2}$. Also, we let $D = \mathbb{D}$ and $\hat{D} = \widehat{\mathbb{D}}$ in the picture.
• Figure 4: Illustration for the event ${\cal F}$ in \ref{['eq:Hdef']}. The solid curves starting from $y$ and $z$ stand for $\gamma^{1}$ and $X^{3}$ respectively. The dotted curve from $z$ to $y$ represents $X^{2}$, while $\gamma^{2}$ is the loop erasure of $X^{2} [0, \sigma]$ defined as in \ref{['eq:gammadef']}.
• Figure 5: Illustration for the setup in Definition \ref{['defnf']}. The thin solid curves starting from the boundary of $B_{i}$ and $B_{j}$ stand for $\mathfrak{a}$ and $\mathfrak{c}$ respectively, while the thin dotted curves starting from the boundary of $B_{i}$ and $B_{j}$ stand for $\mathfrak{b}$ and $\mathfrak{d}$ respectively. The thick curves starting from $x_{\mathfrak{a}}$, $x_{\mathfrak{b}}$ and $x_{\mathfrak{d}}$ represent ${\rm LE} ( R^{\mathfrak{a},0} )$, $R^{\mathfrak{b},\mathfrak{c}}$ and $R^{\mathfrak{d}}$ respectively. For technical reasons the paths $\mathfrak{a}$ through $\mathfrak{d}$ are marked in roman font $a$ through $d$ respectively.

Theorems & Definitions (98)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1
• proof
• Lemma 3.1: Asymptotic Independence
• Lemma 3.2: The Separation Lemma
• Lemma 3.3: Bound on the total variation