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Random walk in cooling random environment: ergodic limits and concentration inequalities

Avena, Luca; Chino, Yuki; da Costa, Conrado; den Hollander, Frank

Random walk in cooling random environment: ergodic limits and concentration inequalities Thumbnail


Authors

Luca Avena

Yuki Chino

Frank den Hollander



Abstract

In previous work by Avena and den Hollander [3], a model of a random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a given sequence of times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure, in dimension one. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over diverging periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moment bounds and concentration estimates. Along the way we derive two results of independent interest, namely, concentration inequalities for the random walk in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the cooling scheme, and by hinting at what can be done in higher dimensions. We argue that, while the cooling scheme does not affect the speed in the strong law of large numbers nor the rate function in the large deviation principle, it does affect the fluctuation properties.

Citation

Avena, L., Chino, Y., da Costa, C., & den Hollander, F. (2019). Random walk in cooling random environment: ergodic limits and concentration inequalities. Electronic Journal of Probability, 24, Article 38. https://doi.org/10.1214/19-ejp296

Journal Article Type Article
Acceptance Date Mar 17, 2019
Online Publication Date Apr 9, 2019
Publication Date Apr 9, 2019
Deposit Date Oct 6, 2019
Publicly Available Date Mar 29, 2024
Journal Electronic Journal of Probability
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 24
Article Number 38
DOI https://doi.org/10.1214/19-ejp296
Related Public URLs https://arxiv.org/abs/1803.03295

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