Cover time for branching random walks on regular trees

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 chose...

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Bibliographic Details
Published in:Journal of applied probability 2022-03, Vol.59 (1), p.256-277
Main Author: Roberts, Matthew I.
Format: Article
Language:English
Subjects:
Children
Heuristic
Original Article
Random numbers
Random variables
Random walk
Trees
Citations: Items that this one cites
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Summary: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 $r + ({2}/{\log(3/2)})\log\log r + {\mathrm{o}}(\log\log r)$ .
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2021.46