## 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 α ∈ (1,2]. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is 2α/(2α – 1). Furthermore, we demonstrate that when α ∈ (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 α = 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.

Original language | English |
---|---|

Pages (from-to) | 1419-1441 |

Number of pages | 23 |

Journal | Electronic Journal of Probability |

Volume | 13 |

DOIs | |

Publication status | Published - 2008 Jan 1 |

Externally published | Yes |

## Keywords

- Branching process
- Random walk
- Stable distribution
- Transition density

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty