Electronic Journal of Probability

On the speed of once-reinforced biased random walk on trees

Andrea Collevecchio, Mark Holmes, and Daniel Kious

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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.

Article information

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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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

random walk once-reinforced random walk Galton-Watson tree reinforcement

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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

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