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Cover time for branching random walks on regular trees

Part of: Geometric probability and stochastic geometry Markov processes

Published online by Cambridge University Press:  09 February 2022

Matthew I. Roberts Open the ORCID record for Matthew I. Roberts [Opens in a new window]
Matthew I. Roberts*
Affiliation:
University of Bath
*
*Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: mattiroberts@gmail.com
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Abstract

Let T be the regular tree in which every vertex has exactly $d\ge 3$ neighbours. Run a branching random walk on T, in which at each time step every particle gives birth to a random number of children with mean d and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex 0 and conditionally on survival of the process, the time it takes for every vertex within distance r of 0 to be hit by a particle of the branching random walk is $r + ({2}/{\log(3/2)})\log\log r + {\mathrm{o}}(\log\log r)$ .

Keywords

Branching random walktreecover timerightmost particle

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 256 - 277
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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