Open Access
2020 UST branches, martingales, and multiple SLE(2)
Alex Karrila
Electron. J. Probab. 25: 1-37 (2020). DOI: 10.1214/20-EJP485


We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple $\mathrm {SLE}(2)$, i.e., an $\mathrm {SLE}(2)$ 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 proof for a boundary-visiting UST branch and a boundary-visiting $\mathrm {SLE}(2)$.


Download Citation

Alex Karrila. "UST branches, martingales, and multiple SLE(2)." Electron. J. Probab. 25 1 - 37, 2020.


Received: 13 March 2020; Accepted: 12 June 2020; Published: 2020
First available in Project Euclid: 18 July 2020

zbMATH: 07252715
MathSciNet: MR4125788
Digital Object Identifier: 10.1214/20-EJP485

Primary: 60Dxx
Secondary: 39A12 , 60G42 , 60J67 , 82B20 , 82B27

Keywords: multiple SLEs , scaling limits , Schramm–Loewner evolutions (SLEs) , uniform spanning tree (UST)

Vol.25 • 2020
Back to Top