Brownian paths as loop-decorated SLEs

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$_2$.

💡 Research Summary

The paper introduces a deterministic operation Ξ that takes a simple curve and a (possibly infinite) collection of loops and produces a new continuous, parametrised path by attaching the loops in the order in which the curve first meets them. The authors carefully analyse three types of singularities that can obstruct continuity—one‑sided intersections, simultaneous first contacts, and loops visited at multiple times—and define a large subset R of configuration pairs (path, loops) that avoids these pathologies. On R they introduce a strong metric d_R₀ and prove that Ξ is continuous with respect to this metric (Theorem 4.11), using auxiliary lemmas that reduce continuity to simpler checks.

The probabilistic core of the work concerns an independent radial SLE₂ curve γ in a simply connected domain D and a Brownian loop soup L of intensity c = 1. Let L_γ be the sub‑collection of loops intersecting γ. By ordering the loops according to the first time γ meets them, the chronological concatenation Ξ(γ, L) yields a continuous path. The main result (Theorem 1.1) shows that this path has exactly the law of a planar Brownian motion started at the origin and stopped upon exiting D, thereby resolving the conjecture of Lawler and Werner (2004).

A substantial part of the paper is devoted to showing that the pair (γ, L) almost surely belongs to R (Proposition 5.1) and that discrete approximations converge to it in the topology induced by d_R₀. The authors construct lattice versions: a simple random walk W_n on the scaled lattice (1/n)ℤ², its loop‑erased version γ_n, and an independent random‑walk loop soup L_n. They prove a strong coupling (Theorem 1.2) such that, with probability at least 1 − ε_n, the supremum norm distance between W_n and the Brownian motion W, as well as the Hausdorff distance between γ_n and γ, are bounded by ε_n. This coupling also extends to the triple (γ_n, L_n, W_n), showing that the discrete construction mirrors the continuous one.

The technical heart lies in controlling the contribution of small loops. The authors introduce the notion of a δ‑isomorphism between the Brownian and random‑walk loop soups and prove that the total time spent on loops below a mesoscopic scale is negligible (Section 6.2). These estimates, together with the continuity of Ξ on R, allow the reduction of the continuous problem to a purely combinatorial one on the lattice.

The paper further demonstrates the robustness of the method. The same arguments apply verbatim to the chordal setting (SLE₂ in the upper half‑plane and a Brownian excursion) and to the massive (off‑critical) regime. In the massive case, each loop in the soup is independently retained with probability exp(−m² t_ℓ), where t_ℓ is its duration. Corollary 1.4 shows that attaching the surviving loops to a massive SLE₂ curve yields a Brownian motion killed at rate m², conditioned to exit the domain before being killed.

Overall, the work provides a complete and rigorous construction of a coupling between SLE₂ and planar Brownian motion via chronological loop addition, settles a long‑standing conjecture, and opens the door to analogous results for other values of κ, other geometries (e.g., Riemann surfaces), and off‑critical models. The blend of deterministic topological analysis with delicate probabilistic estimates makes the approach both powerful and widely applicable.