## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 33, 35 pp.

### Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks

Vladas Sidoravicius and Augusto Teixeira

#### Abstract

We study the dynamics of two conservative lattice gas models on the infinite $d$-dimensional hypercubic lattice: the Activated Random Walks (ARW) and the Stochastic Sandpiles Model (SSM), introduced in the physics literature in the early nineties. Theoretical arguments and numerical analysis predicted that the ARW and SSM undergo a phase transition between an *absorbing phase* and an *active phase* as the initial density crosses a critical threshold. However a rigorous proof of the existence of an absorbing phase was known only for one-dimensional systems. In the present work we establish the existence of such phase transition in any dimension. Moreover, we obtain several quantitative bounds for how fast the activity ceases at a given site or on a finite system. The multi-scale analysis developed here can be extended to other contexts providing an efficient tool to study non-equilibrium phase transitions.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 33, 35 pp.

**Dates**

Received: 4 January 2016

Accepted: 15 March 2017

First available in Project Euclid: 13 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1492070448

**Digital Object Identifier**

doi:10.1214/17-EJP50

**Mathematical Reviews number (MathSciNet)**

MR3646059

**Zentralblatt MATH identifier**

1362.60089

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82C22: Interacting particle systems [See also 60K35] 82C26: Dynamic and nonequilibrium phase transitions (general)

**Keywords**

Particle systems absorption sandpiles non-equilibrium phase transitions

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Sidoravicius, Vladas; Teixeira, Augusto. Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks. Electron. J. Probab. 22 (2017), paper no. 33, 35 pp. doi:10.1214/17-EJP50. https://projecteuclid.org/euclid.ejp/1492070448