## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 11 (2006), paper no. 21, 513-539.

### Existence of multi-dimensional infinite volume self-organized critical forest-fire models

#### Abstract

Consider the following forest-fire model where the possible locations of trees are the sites of a cubic lattice. Each site has two possible states: 'vacant' or 'occupied'. Vacant sites become occupied according to independent rate 1 Poisson processes. Independently, at each site ignition (by lightning) occurs according to independent rate lambda Poisson processes. When a site is ignited, its occupied cluster becomes vacant instantaneously. If the lattice is one-dimensional or finite, then with probability one, at each time the state of a given site only depends on finitely many Poisson events; a process with the above description can be constructed in a standard way. If the lattice is infinite and multi-dimensional, in principle, the state of a given site can be influenced by infinitely many Poisson events in finite time.

#### Article information

**Source**

Electron. J. Probab., Volume 11 (2006), paper no. 21, 513-539.

**Dates**

Accepted: 16 July 2006

First available in Project Euclid: 31 May 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1464730556

**Digital Object Identifier**

doi:10.1214/EJP.v11-333

**Mathematical Reviews number (MathSciNet)**

MR2242654

**Zentralblatt MATH identifier**

1109.60081

**Subjects**

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

Secondary: 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 82C22: Interacting particle systems [See also 60K35]

**Keywords**

forest-fires self-organized criticality forest-fire model existence well-defined

**Rights**

This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Maximilian, Duerre. Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. J. Probab. 11 (2006), paper no. 21, 513--539. doi:10.1214/EJP.v11-333. https://projecteuclid.org/euclid.ejp/1464730556