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

Andrea Collevecchio, Mark Holmes, Daniel Kious

Research output: Contribution to journalArticleResearchpeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Article number86
Number of pages32
JournalElectronic Journal of Probability
Publication statusPublished - 1 Jan 2018


  • Galton-watson tree
  • Once-reinforced random walk
  • Random walk
  • Reinforcement

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