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July 2019 Poly-logarithmic localization for random walks among random obstacles
Jian Ding, Changji Xu
Ann. Probab. 47(4): 2011-2048 (July 2019). DOI: 10.1214/18-AOP1300

Abstract

Place an obstacle with probability $1-\mathsf{p}$ independently at each vertex of $\mathbb{Z}^{d}$, and run a simple random walk until hitting one of the obstacles. For $d\geq2$ and $\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following path localization holds for environments with probability tending to 1 as $n\to\infty$: conditioned on survival up to time $n$ we have that ever since $o(n)$ steps the simple random walk is localized in a region of volume poly-logarithmic in $n$ with probability tending to 1. The previous best result of this type went back to Sznitman (1996) on Brownian motion among Poisson obstacles, where a localization (only for the end point) in a region of volume $t^{o(1)}$ was derived conditioned on the survival of Brownian motion up to time $t$.

Citation

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Jian Ding. Changji Xu. "Poly-logarithmic localization for random walks among random obstacles." Ann. Probab. 47 (4) 2011 - 2048, July 2019. https://doi.org/10.1214/18-AOP1300

Information

Received: 1 September 2017; Revised: 1 July 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114710
MathSciNet: MR3980914
Digital Object Identifier: 10.1214/18-AOP1300

Subjects:
Primary: 60G70, 60H25, 60K37

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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