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Brownian paths as loop-decorated SLEs

Nathanaël Berestycki, Isao Sauzedde

TL;DR

This work proves that attaching loops from a Brownian loop soup to a radial SLE$_2$ path in a planar domain yields a continuous path with the law of planar Brownian motion, thereby coupling SLE$_2$ with Brownian motion and resolving a Lawler–Werner conjecture. The authors construct a deterministic attachment map $\Xi$ on a carefully designed configuration space, establish its time-continuity, and show that the continuum limit of lattice models (LERW with RW loop soups) recovers the Brownian path as the loop-augmented SLE$_2$ trace is traversed. They provide a robust framework that also extends to near-critical (massive) SLE$_2$ limits and to the chordal setting, with precise lattice-to-continuum convergence results. The results give a natural probabilistic interpretation of SLE$_2$ as a loop-erasure of Brownian motion, and they establish a scaling-limit link between loop-erased random walk and ordinary random walk with loop soups, yielding potential applicability to off-critical regimes. Overall, the paper delivers a rigorous pathwise coupling, a versatile attachment mechanism, and a solid bridge between discrete and continuum conformally invariant processes with significant implications for scaling limits in two dimensions.

Abstract

We construct an application, which takes as input a simple path and a possibly infinite collection of loops, and outputs a continuous path by adding the loops chronologically to the simple path as the simple path encounters them. By studying the regularity properties of this application and using lattice discretisations, we prove that chronologically adding the loops from a Brownian loop soup encountered by an independent radial SLE$_2$ path produces a continuous path which has the law of a planar Brownian motion. This resolves a conjecture of Lawler and Werner. This construction produces a coupling between SLE$_2$ and Brownian motion, and we further show that this joint law is the scaling limit of the loop-erased random walk and the random walk itself. The arguments are robust and can be applied for instance in the off-critical setup, where the scaling limit of loop-erased random walk is Makarov and Smirnov's massive SLE$_2$.

Brownian paths as loop-decorated SLEs

TL;DR

This work proves that attaching loops from a Brownian loop soup to a radial SLE path in a planar domain yields a continuous path with the law of planar Brownian motion, thereby coupling SLE with Brownian motion and resolving a Lawler–Werner conjecture. The authors construct a deterministic attachment map on a carefully designed configuration space, establish its time-continuity, and show that the continuum limit of lattice models (LERW with RW loop soups) recovers the Brownian path as the loop-augmented SLE trace is traversed. They provide a robust framework that also extends to near-critical (massive) SLE limits and to the chordal setting, with precise lattice-to-continuum convergence results. The results give a natural probabilistic interpretation of SLE as a loop-erasure of Brownian motion, and they establish a scaling-limit link between loop-erased random walk and ordinary random walk with loop soups, yielding potential applicability to off-critical regimes. Overall, the paper delivers a rigorous pathwise coupling, a versatile attachment mechanism, and a solid bridge between discrete and continuum conformally invariant processes with significant implications for scaling limits in two dimensions.

Abstract

We construct an application, which takes as input a simple path and a possibly infinite collection of loops, and outputs a continuous path by adding the loops chronologically to the simple path as the simple path encounters them. By studying the regularity properties of this application and using lattice discretisations, we prove that chronologically adding the loops from a Brownian loop soup encountered by an independent radial SLE path produces a continuous path which has the law of a planar Brownian motion. This resolves a conjecture of Lawler and Werner. This construction produces a coupling between SLE and Brownian motion, and we further show that this joint law is the scaling limit of the loop-erased random walk and the random walk itself. The arguments are robust and can be applied for instance in the off-critical setup, where the scaling limit of loop-erased random walk is Makarov and Smirnov's massive SLE.
Paper Structure (32 sections, 53 theorems, 193 equations, 3 figures)

This paper contains 32 sections, 53 theorems, 193 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Precise statements of Theorem \ref{['thm:main_intro']}
  4. Strategy and organisation of the paper
  5. Construction of the path Xi(gamma,L)
  6. Preliminary notations
  7. Summary of the construction
  8. The sets R,R0
  9. Tie-breaks
  10. The set T and the function sigma
  11. Continuity in time of Xi(gamma,L,lambda,B)
  12. Attaching a single loop to a simple path
  13. The set of regular pairs
  14. L0 as a neighborhood of Lreg
  15. Continuity on Lreg
  16. ...and 17 more sections

Key Result

Theorem 1

In distribution, $\mathop{\mathrm{Range}}\nolimits( W)=\mathop{\mathrm{Range}}\nolimits(\gamma)\cup \bigcup_{\ell \in \mathcal{L}_\gamma } \mathop{\mathrm{Range}}\nolimits(\ell).$

Figures (3)

  • Figure 1: The three types of issues that prevent continuity of the map ${\Xi}$. In red, the simple path. In black and blue, loops $\ell,\ell'\in \mathcal{L}$. Around the root point $x_\ell$ where $\gamma$ meets $\ell$ for the first time: Left: the intersection is one-sided. Small continuous deformations of $\gamma$ or $\ell$ can remove this intersection entirely, rather than moving it continuously. Middle:$\gamma$ also meets $\ell'$ for the first time at the same place as $\ell$. Small continuous deformations can make $\gamma$ meet either $\ell$ or $\ell'$ first. The resulting path are very different, although their range are about the same. Right:$\ell$ passes through $x_\ell$ at two distinct times $s,t$. Some small deformations lead to rerooting $\ell$ around $s$, while others lead to rerooting $\ell$ around $t$, again leading to two fairly distinct paths.
  • Figure 2: Three examples of bad events. In red (thick lines), the $SLE_2$ process. In black (thin lines), one of the loop in $\mathcal{L}$. Dashed lines: their discrete counterparts. left: Although $\ell$ and $\gamma$ do not intersect, the corresponding discrete processes intersect, so the reconstructed path $W(\mathbb{L}, \mathsf{S})$ will incorporate this loop. middle: The opposite event. right: The paths $\ell$ and $\gamma$ intersect, so does their discrete equivalent, the continuum and discrete roots (bullet and squared bullet) are close from each other spatially, but correspond to very different parts of $\gamma$. Remark it can also happen that the continuum and discrete roots both exist but are spatially far from each other. If the red curves now represent the boundary $\partial D$ rather than the $SLE_2$ process, these figures depict other atypical events that we need control over.
  • Figure 3: The fatter parts of the trajectory ($LE(S)_{[0,s]}$ in red, $S_{[\sigma_{M_0}, \tau ]}$ in blue) do not intersect each other.

Theorems & Definitions (108)

  • Theorem : SapozhnikovShiraishi
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 98 more