## The Annals of Probability

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

#### 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2121-2133.

Dates
Revised: June 2017
First available in Project Euclid: 13 June 2018

https://projecteuclid.org/euclid.aop/1528876823

Digital Object Identifier
doi:10.1214/17-AOP1222

Mathematical Reviews number (MathSciNet)
MR3813987

Zentralblatt MATH identifier
06919020

#### Citation

Kious, Daniel; Sidoravicius, Vladas. Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees. Ann. Probab. 46 (2018), no. 4, 2121--2133. doi:10.1214/17-AOP1222. https://projecteuclid.org/euclid.aop/1528876823

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