## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 23 (2018), paper no. 43, 11 pp.

### Random walk on the randomly-oriented Manhattan lattice

Sean Ledger, Bálint Tóth, and Benedek Valkó

#### Abstract

In the randomly-oriented Manhattan lattice, every line in $\mathbb{Z} ^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $\mathbb{Z} ^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.

#### Article information

**Source**

Electron. Commun. Probab., Volume 23 (2018), paper no. 43, 11 pp.

**Dates**

Received: 5 February 2018

Accepted: 19 June 2018

First available in Project Euclid: 25 July 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1532505674

**Digital Object Identifier**

doi:10.1214/18-ECP144

**Mathematical Reviews number (MathSciNet)**

MR3841404

**Zentralblatt MATH identifier**

1397.82046

**Subjects**

Primary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

**Keywords**

Random walks in random environment superdiffusivity

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Ledger, Sean; Tóth, Bálint; Valkó, Benedek. Random walk on the randomly-oriented Manhattan lattice. Electron. Commun. Probab. 23 (2018), paper no. 43, 11 pp. doi:10.1214/18-ECP144. https://projecteuclid.org/euclid.ecp/1532505674