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

Avena, Luca and Chino, Yuki and da Costa, Conrado and den Hollander, Frank (2019) 'Random walk in cooling random environment : ergodic limits and concentration inequalities.', Electronic journal of probability., 24 . p. 38.


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.

Item Type:Article
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Publisher statement:This paper has been published under a Creative Commons Attribution 4.0 International License.
Date accepted:17 March 2019
Date deposited:15 November 2019
Date of first online publication:09 April 2019
Date first made open access:15 November 2019

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