Abstract
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.
Citation
Daniel Kious. Vladas Sidoravicius. "Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees." Ann. Probab. 46 (4) 2121 - 2133, July 2018. https://doi.org/10.1214/17-AOP1222
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