Electronic Journal of Probability
- Electron. J. Probab.
- Volume 23 (2018), paper no. 86, 32 pp.
On the speed of once-reinforced biased random walk on trees
We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.
Electron. J. Probab., Volume 23 (2018), paper no. 86, 32 pp.
Received: 6 September 2017
Accepted: 27 July 2018
First available in Project Euclid: 12 September 2018
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Collevecchio, Andrea; Holmes, Mark; Kious, Daniel. On the speed of once-reinforced biased random walk on trees. Electron. J. Probab. 23 (2018), paper no. 86, 32 pp. doi:10.1214/18-EJP208. https://projecteuclid.org/euclid.ejp/1536717745