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2008 Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive
David Croydon, Takashi Kumagai
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Electron. J. Probab. 13: 1419-1441 (2008). DOI: 10.1214/EJP.v13-536

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 $n$.

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David Croydon. Takashi Kumagai. "Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive." Electron. J. Probab. 13 1419 - 1441, 2008. https://doi.org/10.1214/EJP.v13-536

Information

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

zbMATH: 1191.60121
MathSciNet: MR2438812
Digital Object Identifier: 10.1214/EJP.v13-536

Subjects:
Primary: 60K37
Secondary: 60J35, 60J80

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