Electronic Journal of Probability

Cluster growth in the dynamical Erdős-Rényi process with forest fires

Edward Crane, Nic Freeman, and Bálint Tóth

Full-text: Open access

Abstract

We investigate the growth of clusters within the forest fire model of Ráth and Tóth [EJP, vol 14, paper no 45]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Rényi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear.

We give a precise description of the process $C_t$ of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that $C_t$ is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to 1 on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process $C_t$.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 101, 33 pp.

Dates
Accepted: 24 September 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067207

Digital Object Identifier
doi:10.1214/EJP.v20-4035

Mathematical Reviews number (MathSciNet)
MR3407218

Zentralblatt MATH identifier
1328.60208

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C80: Random graphs [See also 60B20] 35Q82: PDEs in connection with statistical mechanics 82C27: Dynamic critical phenomena

Keywords
Erdős-Rényi random graph forest fire self-organized criticality Smoluchowski coagulation equation

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Crane, Edward; Freeman, Nic; Tóth, Bálint. Cluster growth in the dynamical Erdős-Rényi process with forest fires. Electron. J. Probab. 20 (2015), paper no. 101, 33 pp. doi:10.1214/EJP.v20-4035. https://projecteuclid.org/euclid.ejp/1465067207


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