@article{EJP536,
author = {David Croydon and Takashi Kumagai},
title = {Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {13},
year = {2008},
keywords = {random walk; branching process; stable distribution; transition density},
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 $n$.},
pages = {no. 51, 1419-1441},
issn = {1083-6489},
doi = {10.1214/EJP.v13-536},
url = {http://ejp.ejpecp.org/article/view/536}}