Electronic Communications in Probability

Critical percolation and the incipient infinite cluster on Galton-Watson trees

Marcus Michelen

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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


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