15 July 2021 Convergence of loop-erased random walk in the natural parameterization
Gregory F. Lawler, Fredrik Viklund
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Duke Math. J. 170(10): 2289-2370 (15 July 2021). DOI: 10.1215/00127094-2020-0075


Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to SLE2 parameterized by 54-dimensional Minkowski content. In doing this, we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest. We give, for example, two-point estimates, estimates on maximal content, and a “separation lemma” for two-sided LERW.


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Gregory F. Lawler. Fredrik Viklund. "Convergence of loop-erased random walk in the natural parameterization." Duke Math. J. 170 (10) 2289 - 2370, 15 July 2021. https://doi.org/10.1215/00127094-2020-0075


Received: 13 March 2019; Revised: 2 June 2020; Published: 15 July 2021
First available in Project Euclid: 25 May 2021

MathSciNet: MR4291423
zbMATH: 1491.60148
Digital Object Identifier: 10.1215/00127094-2020-0075

Primary: 60J67
Secondary: 60K35

Keywords: conformal invariance , Loop-erased random walk , Schramm–Loewner evolution

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 10 • 15 July 2021
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