## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 15, 11 pp.

### Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

Tal Orenshtein and Renato Soares dos Santos

#### Abstract

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 15, 11 pp.

**Dates**

Received: 13 July 2015

Accepted: 26 January 2016

First available in Project Euclid: 19 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1455897057

**Digital Object Identifier**

doi:10.1214/16-ECP4426

**Mathematical Reviews number (MathSciNet)**

MR3485384

**Zentralblatt MATH identifier**

1338.60102

**Subjects**

Primary: 60F20: Zero-one laws 60K37: Processes in random environments

Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82C44: Dynamics of disordered systems (random Ising systems, etc.)

**Keywords**

random walk dynamic random environment zero-one law transience recurrence

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Orenshtein, Tal; Soares dos Santos, Renato. Zero-one law for directional transience of one-dimensional random walks in dynamic random environments. Electron. Commun. Probab. 21 (2016), paper no. 15, 11 pp. doi:10.1214/16-ECP4426. https://projecteuclid.org/euclid.ecp/1455897057