Electronic Journal of Probability

Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive

David Croydon and Takashi Kumagai

Full-text: Open access

Abstract

We establish a variety of properties of the discrete time simple random walk on a Galton-Watson tree conditioned to survive when the offspring distribution, $Z$ say, is in the domain of attraction of a stable law with index $\alpha\in(1,2]$. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is $2\alpha/(2\alpha-1)$. Furthermore, we demonstrate that when $\alpha\in(1,2)$ there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when $\alpha=2$. In the course of our arguments, we obtain tail bounds for the distribution of the $n$th generation size of a Galton-Watson branching process with offspring distribution $Z$ conditioned to survive, as well as tail bounds for the distribution of the total number of individuals born up to the $n$th generation, that are uniform in $n$.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 51, 1419-1441.

Dates
Accepted: 28 August 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819124

Digital Object Identifier
doi:10.1214/EJP.v13-536

Mathematical Reviews number (MathSciNet)
MR2438812

Zentralblatt MATH identifier
1191.60121

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Keywords
random walk branching process stable distribution transition density

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

Citation

Croydon, David; Kumagai, Takashi. Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electron. J. Probab. 13 (2008), paper no. 51, 1419--1441. doi:10.1214/EJP.v13-536. https://projecteuclid.org/euclid.ejp/1464819124


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