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


 
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1. Title Title of document Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive
 
2. Creator Author's name, affiliation, country David A. Croydon; University of Warwick
 
2. Creator Author's name, affiliation, country Takashi Kumagai; Kyoto University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random walk; branching process; stable distribution; transition density
 
3. Subject Subject classification Primary 60K37; Secondary 60J80, 60J35.
 
4. Description 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 $n$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2008-08-28
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/536
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-536
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 13
 
12. Language English=en
 
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