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Cover time for branching random walks on regular trees

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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… Click to show full abstract

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; time; time branching; cover time; log log

Journal Title: Journal of Applied Probability
Year Published: 2022

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