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November 2019 Four-dimensional loop-erased random walk
Gregory Lawler, Xin Sun, Wei Wu
Ann. Probab. 47(6): 3866-3910 (November 2019). DOI: 10.1214/19-AOP1349

Abstract

The loop-erased random walk (LERW) in $\mathbb{Z}^{4}$ is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost surely and in $L^{p}$ for all $p>0$. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a $\pm 1$ spin model coupled with the wired spanning forests on $\mathbb{Z}^{4}$ with the bi-Laplacian Gaussian field on $\mathbb{R}^{4}$ as its scaling limit.

Citation

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Gregory Lawler. Xin Sun. Wei Wu. "Four-dimensional loop-erased random walk." Ann. Probab. 47 (6) 3866 - 3910, November 2019. https://doi.org/10.1214/19-AOP1349

Information

Received: 1 April 2017; Revised: 1 February 2019; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212173
MathSciNet: MR4038044
Digital Object Identifier: 10.1214/19-AOP1349

Subjects:
Primary: 60G50, 60K35

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • November 2019
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