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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: |
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| 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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