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

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arxiv 1702.01982 v2 pith:A6XVF2UU submitted 2017-02-07 math.PR

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

classification math.PR
keywords randomwalkspeedtreesbiasedorbrwproveunderlying
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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.

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