UST branches, martingales, and multiple SLE(2)

Abstract

We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple SLE(2), i.e., an SLE(2) process weighted by a suitable partition function. By recent results, this also characterizes the "global" scaling limit of the full collection of full curves. The identification is based on a martingale observable in the UST with N branches, obtained by weighting the well-known martingale in the UST with one branch by the discrete partition functions of the models. The obtained weighting transforms of the discrete martingales and the limiting SLE processes, respectively, only rely on a discrete domain Markov property and (essentially) the convergence of partition functions. We illustrate their generalizability by sketching an analogous convergence result for a boundary-visiting UST branch and boundary-visiting SLE(2).

Hypergeometric SLE with $κ=8$: Convergence of UST and LERW in Topological Rectangles

“…We define T δ M and τ δ ,z similarly for ηL δ . We may assume T δ M → T M , τ δ ,z → τ ,z and τ δ → τ by considering a continuous modification, see details in [Kar19] and [Kar20]. Then, Lemma 4.6 implies that…”

“…We may assume T δ → T almost surely as δ → 0 by considering the continuous modification, see details in [Kar19] and [Kar20].…”

On multiple SLE for the FK-Ising model

“…. , 2n} without fixed points [36,37,35]. The structure of the partition function is related to the determinantal nature of the UST and the Fomin identity [23].…”

On the convergence of massive loop-erased random walks to massive SLE(2) curves

“…Dmitry Chelkak is grateful to Stanislav Smirnov for explaining the ideas of [23] during several conversations dating back to 2009-2014. We want to thank Michel Bauer, Konstantin Izyurov and Kalle Kytölä for valuable comments and for encouraging us to write this paper; Alex Karrila for useful discussions of his research [12,13]; Chengyang Shao for discussions during his spring 2017 internship at the ENS [27]; and Mikhail Skopenkov for a feedback, which in particular included pointing out a mess at the end of the proof of [7,Theorem 3.13]. This proof is sketched in Section 3 with a necessary correction.…”