Croydon, David A. and Kumagai, Takashi (2008) Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electronic Journal of Probability, Vol.13 . pp. 1419-1441.

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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 is an element of (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 is an element of (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 nth 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 nth generation, that are uniform in n.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Statistics
Journal or Publication Title:	Electronic Journal of Probability
Publisher:	University of Washington. Dept. of Mathematics
ISSN:	1083-6489
Official Date:	28 August 2008
Dates:	Date Event 28 August 2008 Published
Volume:	Vol.13
Number of Pages:	23
Page Range:	pp. 1419-1441
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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