Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 24 (2019), paper no. 10, 13 pp.
Critical percolation and the incipient infinite cluster on Galton-Watson trees
Abstract
We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.
Article information
Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 10, 13 pp.
Dates
Received: 6 June 2018
Accepted: 3 February 2019
First available in Project Euclid: 18 February 2019
Permanent link to this document
https://projecteuclid.org/euclid.ecp/1550480494
Digital Object Identifier
doi:10.1214/19-ECP216
Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Keywords
critical percolation incipient infinite cluster exponential limit law Kolmogorov’s estimate
Rights
Creative Commons Attribution 4.0 International License.
Citation
Michelen, Marcus. Critical percolation and the incipient infinite cluster on Galton-Watson trees. Electron. Commun. Probab. 24 (2019), paper no. 10, 13 pp. doi:10.1214/19-ECP216. https://projecteuclid.org/euclid.ecp/1550480494