## Electronic Journal of Probability

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

#### 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
Accepted: 17 March 2019
First available in Project Euclid: 9 April 2019

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

Digital Object Identifier
doi:10.1214/19-EJP296

Mathematical Reviews number (MathSciNet)
MR3940768

Zentralblatt MATH identifier
07055676

#### 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

#### References

• [1] S. Ahn and J. Peterson, Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments, Bernoulli 25 (2019), no. 2, 1386–1411.
• [2] L. Avena, O. Blondel, and A. Faggionato, Analysis of random walks in dynamic random environments via $L^2$-perturbations, Stoch. Proc. Appl. 128 (2018), no. 10, 3490 – 3530.
• [3] L. Avena and F. den Hollander, Random walks in cooling random environments, arXiv e-prints (2017), arXiv:1610.00641.
• [4] C. Boldrighini, R. Minlos, and A. Pellegrinotti, Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive, Probab. Theory Relat. Fields 129 (2004), no. 1, 133–156.
• [5] D. Campos, A. Drewitz, A. Ramírez, F. Rassoul-Agha, and T. Seppäläinen, Level 1 quenched large deviation principle for random walk in dynamic, Bull. Inst. Math. Acad. Sin. 8 (2013), 1–29.
• [6] F. Comets, N. Gantert, and O. Zeitouni, Quenched, annealed and functional large deviations for one-dimensional random walk in random environment, Probab. Theory Relat. Fields 118 (2000), no. 1, 65–114.
• [7] A. Dembo, Y. Peres, and O. Zeitouni, Tail estimates for one-dimensional random walk in random environment, Commun. in Math. Phys. 181 (1996), no. 3, 667–683.
• [8] A. Greven and F. den Hollander, Large deviations for a random walk in random environment, Ann. Probab. 22 (1994), no. 3, 1381–1428.
• [9] F. den Hollander, Large deviations, Fields Institute monographs, American Mathematical Society, 2000.
• [10] H. Kesten, The limit distribution of Sinai’s random walk in random environment, Phys. A 138 (1986), 299–309.
• [11] H. Kesten, M. Kozlov, F. Spitzer, A limit law for random walk in a random environment, Comp. Math. 30 (1975), no. 2, 145–168.
• [12] T. Liggett, An improved subadditive ergodic theorem, Ann. Probab. 13 (1985), no. 4, 1279–1285.
• [13] P. Révész, The Laws of Large Numbers, Probability and Mathematical Statistics, Academic Press, 1968.
• [14] Y. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl. 27 (1982), no. 2, 256–268.
• [15] F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), no. 1, 1–31.
• [16] O. Zeitouni, Lectures on probability theory and statistics, Lecture Notes in Mathematics, vol. 1837, 189–312, 2004.