Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive

David Croydon, Takashi Kumagai

研究成果: Article査読

21 被引用数 (Scopus)

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
ページ(範囲)1419-1441
ページ数23
ジャーナルElectronic Journal of Probability
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

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