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

David Croydon, Takashi Kumagai

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)

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 languageEnglish
Pages (from-to)1419-1441
Number of pages23
JournalElectronic Journal of Probability
Volume13
DOIs
Publication statusPublished - 2008 Jan 1
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint

Dive into the research topics of 'Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive'. Together they form a unique fingerprint.

Cite this