## The Annals of Probability

### Poly-logarithmic localization for random walks among random obstacles

#### 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 $t$.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2011-2048.

Dates
Revised: July 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205700

Digital Object Identifier
doi:10.1214/18-AOP1300

Mathematical Reviews number (MathSciNet)
MR3980914

Zentralblatt MATH identifier
07114710

#### Citation

Ding, Jian; Xu, Changji. Poly-logarithmic localization for random walks among random obstacles. Ann. Probab. 47 (2019), no. 4, 2011--2048. doi:10.1214/18-AOP1300. https://projecteuclid.org/euclid.aop/1562205700

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