Cover time for branching random walks on regular trees

Research output: Contribution to journalArticlepeer-review


Let T be the regular tree in which every vertex has exactly <![CDATA[ $d\ge 3$ ]]> 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 <![CDATA[ $r + ({2}/{\log(3/2)})\log\log r + {\mathrm{o}}(\log\log r)$ ]]>.

Original languageEnglish
Pages (from-to)256-277
JournalJournal of Applied Probability
Issue number1
Early online date9 Feb 2022
Publication statusPublished - 31 Mar 2022

Bibliographical note

Funding Information:
This work was supported by a Royal Society University Research Fellowship.

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust.


  • Branching random walk
  • cover time
  • rightmost particle
  • tree

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


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