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2021 On the convergence of massive loop-erased random walks to massive SLE(2) curves
Dmitry Chelkak, Yijun Wan
Author Affiliations +
Electron. J. Probab. 26: 1-35 (2021). DOI: 10.1214/21-EJP615

Abstract

Following the strategy proposed by Makarov and Smirnov [24] in 2009 (see also [3, 2] for theoretical physics arguments), we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up to [24] appeared since then, we believe that such a treatment might be of interest to the community. We do not require any regularity of the limiting planar domain near its degenerate prime ends a and b except that (Ωδ,aδ,bδ) are assumed to be ‘close discrete approximations’ to (Ω,a,b) near a and b in the sense of [13].

Funding Statement

Supported by the ENS–MHI chair funded by MHI and by the ANR-18-CE40-0033 project DIMERS.

Acknowledgments

Dmitry Chelkak is grateful to Stanislav Smirnov for explaining the ideas of [24] 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 [13, 14]; Chengyang Shao for discussions during his spring 2017 internship at the ENS [31]; and Mikhail Skopenkov for a feedback, which in particular included pointing out a mess at the end of the proof of [8, Theorem 3.13]. This proof is sketched in Section 3 with a necessary correction. Last but not least we want to thank the referees for their careful reading of the paper and for useful comments and suggestions on the presentation of the material.

Citation

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Dmitry Chelkak. Yijun Wan. "On the convergence of massive loop-erased random walks to massive SLE(2) curves." Electron. J. Probab. 26 1 - 35, 2021. https://doi.org/10.1214/21-EJP615

Information

Received: 19 February 2020; Accepted: 29 March 2021; Published: 2021
First available in Project Euclid: 3 May 2021

Digital Object Identifier: 10.1214/21-EJP615

Subjects:
Primary: 60Dxx, 82B20

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