Electronic Journal of Probability

New examples of ballistic RWRE in the low disorder regime

Alejandro F. Ramírez and Santiago Saglietti

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We give a new criterion for ballistic behavior of random walks in random environments which are low disorder perturbations of the simple symmetric random walk on $\mathbb{Z} ^{d}$, for $d\geq 2$. This extends the results from 2003 established by Sznitman in [12] and, in particular, allow us to give new examples of ballistic RWREs in dimension $d=3$ which do not satisfy Kalikow’s condition, through a new sharp version of Kalikow’s criteria. Essentially, this new criterion states that ballisticity occurs whenever the average local drift of the walk is not too small when compared to the standard deviation of the environment. Its proof relies on applying coarse-graining methods together with a variation of the Azuma-Hoeffding concentration inequality in order to verify the fulfillment of a ballisticity condition by Berger, Drewitz and Ramírez.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 127, 20 pp.

Received: 28 April 2019
Accepted: 16 October 2019
First available in Project Euclid: 9 November 2019

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Primary: 60K37: Processes in random environments 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

random walk in random environment small perturbations of simple random walk ballistic behavior concentration inequalities

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Ramírez, Alejandro F.; Saglietti, Santiago. New examples of ballistic RWRE in the low disorder regime. Electron. J. Probab. 24 (2019), paper no. 127, 20 pp. doi:10.1214/19-EJP374. https://projecteuclid.org/euclid.ejp/1573268591

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