## Electronic Communications in Probability

### Functional weak limit of random walks in cooling random environment

Yongjia Xie

#### Abstract

We prove an annealed weak limit of the trajectory of the random walks in cooling random environment (RWCRE) under both slow (polynomial) and fast (exponential) cooling. We identify the weak limit when the underlying static environment is recurrent (Sinai’s model). Avena and den Hollander have previously proved a Gaussian limiting distribution for the distribution of the endpoint of the walk. We find that the weak limit of the trajectory exists as a time-rescaled Brownian motion in the slow cooling case but the limit degenerates to a constant function in the fast cooling one.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 86, 14 pp.

Dates
Accepted: 6 November 2020
First available in Project Euclid: 30 December 2020

https://projecteuclid.org/euclid.ecp/1609319091

Digital Object Identifier
doi:10.1214/20-ECP360

#### Citation

Xie, Yongjia. Functional weak limit of random walks in cooling random environment. Electron. Commun. Probab. 25 (2020), paper no. 86, 14 pp. doi:10.1214/20-ECP360. https://projecteuclid.org/euclid.ecp/1609319091

#### References

• [1] L. Avena, F. den Hollander, and F. Redig, Law of large numbers for a class of random walks in dynamic random environments, Electron. J. Probab. 16 (2011), no. 21, 587–617.
• [2] Luca Avena, Yuki Chino, Conrado da Costa, and Frank den Hollander, Random walk in cooling random environment: ergodic limits and concentration inequalities, Electron. J. Probab. 24 (2019), Paper No. 38, 35.
• [3] Luca Avena, Yuki Chino, Conrado da Costa, and Frank den Hollander, Random walk in cooling random environment: recurrence versus transience and mixed fluctuations, 2019. arXiv:1903.09200v2
• [4] Luca Avena, Renato Soares dos Santos, and Florian Völlering, Transient random walk in symmetric exclusion: limit theorems and an Einstein relation, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 2, 693–709.
• [5] Luca Avena and Frank den Hollander, Random walks in cooling random environments, Sojourns in Probability Theory and Statistical Physics – III (Singapore) (Vladas Sidoravicius, ed.), Springer Singapore, 2019, pp. 23–42.
• [6] Patrick Billingsley, Convergence of probability measures, second ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.
• [7] Oriane Blondel, Marcelo R. Hilário, and Augusto Teixeira, Random walks on dynamical random environments with nonuniform mixing, Ann. Probab. 48 (2020), no. 4, 2014–2051.
• [8] F. den Hollander and R. S. dos Santos, Scaling of a random walk on a supercritical contact process, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 4, 1276–1300.
• [9] Frank den Hollander, Harry Kesten, and Vladas Sidoravicius, Random walk in a high density dynamic random environment, Indag. Math. (N.S.) 25 (2014), no. 4, 785–799.
• [10] Rick Durrett, Probability—theory and examples, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 49, Cambridge University Press, Cambridge, 2019, Fifth edition of [MR1068527].
• [11] M. R. Hilário, F. den Hollander, R. S. dos Santos, V. Sidoravicius, and A. Teixeira, Random walk on random walks, Electron. J. Probab. 20 (2015), no. 95, 35.
• [12] Marcelo R. Hilário, Daniel Kious, and Augusto Teixeira, Random Walk on the Simple Symmetric Exclusion Process, Comm. Math. Phys. 379 (2020), no. 1, 61–101.
• [13] François Huveneers and François Simenhaus, Random walk driven by the simple exclusion process, Electron. J. Probab. 20 (2015), no. 105, 42.
• [14] H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145–168.
• [15] Harry Kesten, The limit distribution of Sinaĭ’s random walk in random environment, Phys. A 138 (1986), no. 1-2, 299–309.
• [16] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York-Heidelberg, 1975, Translated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.
• [17] Ya. G. Sinai, The limit behavior of a one-dimensional random walk in a random environment, Teor. Veroyatnost. i Primenen. 27 (1982), no. 2, 247–258.
• [18] Fred Solomon, Random walks in a random environment, Ann. Probability 3 (1975), 1–31.
• [19] Ofer Zeitouni, Random walks in random environment, Lectures on probability theory and statistics, Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004, pp. 189–312.