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

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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. ISSN 1083-6489.

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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:
DateEvent
28 August 2008Published
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
Date of first compliant deposit: 14 December 2015
Date of first compliant Open Access: 14 December 2015

Data sourced from Thomson Reuters' Web of Knowledge

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