Annals of Probability

A lower bound for disconnection by simple random walk

Xinyi Li

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We consider simple random walk on $\mathbb{Z}^{d}$, $d\geq3$. Motivated by the work of A.-S. Sznitman and the author in [Probab. Theory Related Fields 161 (2015) 309–350] and [Electron. J. Probab. 19 (2014) 1–26], we investigate the asymptotic behavior of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk. We derive asymptotic lower bounds that bring into play random interlacements. Although open at the moment, some of the lower bounds we obtain possibly match the asymptotic upper bounds recently obtained in [Disconnection, random walks, and random interlacements (2014)]. This potentially yields special significance to the tilted walks that we use in this work, and to the strategy that we employ to implement disconnection.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 879-931.

Received: January 2015
Revised: September 2015
First available in Project Euclid: 31 March 2017

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Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 82B43: Percolation [See also 60K35]

Large deviations random walk random interlacements


Li, Xinyi. A lower bound for disconnection by simple random walk. Ann. Probab. 45 (2017), no. 2, 879--931. doi:10.1214/15-AOP1077.

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