## 抄録

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.

本文言語 | English |
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ページ（範囲） | 1419-1441 |

ページ数 | 23 |

ジャーナル | Electronic Journal of Probability |

巻 | 13 |

DOI | |

出版ステータス | Published - 2008 1月 1 |

外部発表 | はい |

## ASJC Scopus subject areas

- 統計学および確率
- 統計学、確率および不確実性