# Cover time for branching random walks on regular trees

## TL;DR

This paper analyzes the cover time of a branching random walk on a regular tree, showing that the time to hit all vertices within distance r grows approximately as r plus a logarithmic correction term.

## Contribution

It provides a precise asymptotic estimate for the cover time of branching random walks on regular trees, including a second-order logarithmic correction.

## Key findings

- Cover time is almost surely r + (2 / log(3/2)) * log log r + o(log log r)
- Asymptotic behavior holds conditioned on survival of the process
- Results extend understanding of branching processes on trees

## 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 almost surely $r + \frac{2}{\log(3/2)}\log\log r + o(\log\log r)$.