Electronic Journal of Probability
- Electron. J. Probab.
- Volume 20 (2015), paper no. 101, 33 pp.
Cluster growth in the dynamical Erdős-Rényi process with forest fires
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 $C_t$.
Electron. J. Probab., Volume 20 (2015), paper no. 101, 33 pp.
Accepted: 24 September 2015
First available in Project Euclid: 4 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
This work is licensed under aCreative Commons Attribution 3.0 License.
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