Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees

@article{Kious2018PhaseTF,
  title={Phase transition for the Once-reinforced random walk on \$\mathbb\{Z\}^\{d\}\$-like trees},
  author={Daniel Kious and Vladas Sidoravicius},
  journal={The Annals of Probability},
  year={2018}
}
In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_0$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_0$ and almost surely transient if $a<a_0$. This provides the first examples of phase transition for the Once-reinforced random walk. 

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