## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 24 (2019), paper no. 38, 35 pp.

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

Luca Avena, Yuki Chino, Conrado da Costa, and 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.

#### Article information

**Source**

Electron. J. Probab., Volume 24 (2019), paper no. 38, 35 pp.

**Dates**

Received: 25 October 2018

Accepted: 17 March 2019

First available in Project Euclid: 9 April 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1554775418

**Digital Object Identifier**

doi:10.1214/19-EJP296

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60F10: Large deviations 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

**Keywords**

random walk dynamic random environment resampling times law of large numbers large deviation principle concentration inequalities

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Avena, Luca; Chino, Yuki; da Costa, Conrado; den Hollander, Frank. Random walk in cooling random environment: ergodic limits and concentration inequalities. Electron. J. Probab. 24 (2019), paper no. 38, 35 pp. doi:10.1214/19-EJP296. https://projecteuclid.org/euclid.ejp/1554775418