## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 21, Number (2016), paper no. 13, 15 pp.

### Absorbing-state phase transition in biased activated random walk

#### Abstract

We consider the activated random walk model on $\mathbb{Z} ^d$, which undergoes a transition from an absorbing regime to a regime of sustained activity. A central question for this model involves the estimation of the critical density $\mu _c$. We prove that if the jump distribution is biased, then $\mu _c < 1$ for any sleeping rate $\lambda $, $d \geq 1$, and that $\mu _c \to 0$ as $\lambda \to 0$ in one dimension. This answers a question from *Rolla and Sidoravicius* (2012) and *Dickman, Rolla and Sidoravicius* (2010) in the case of biased jump distribution. Furthermore, we prove that the critical density depends on the jump distribution.

#### Article information

**Source**

Electron. J. Probab. Volume 21, Number (2016), paper no. 13, 15 pp.

**Dates**

Received: 30 April 2015

Accepted: 27 January 2016

First available in Project Euclid: 23 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1456246244

**Digital Object Identifier**

doi:10.1214/16-EJP4275

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]

**Keywords**

interacting particles activated random walk lattice gasses Abelian networks

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Taggi, Lorenzo. Absorbing-state phase transition in biased activated random walk. Electron. J. Probab. 21 (2016), paper no. 13, 15 pp. doi:10.1214/16-EJP4275. https://projecteuclid.org/euclid.ejp/1456246244.