Abstract
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.
Highlights
Reinforced random walks have been studied extensively since the introduction of the-reinforced random walk of Coppersmith and Diaconis [6]
In this paper we study once-reinforced random walk, which was introduced by Davis [7] as a possible simpler model of reinforcement to understand
Biased random walk corresponds to the case u0 = u1 = 1, while unbiased random walk corresponds to the case u0 = u1 = 1
Summary
Reinforced random walks have been studied extensively since the introduction of the (linearly)-reinforced random walk of Coppersmith and Diaconis [6]. The once-reinforced biased random walk on G is a perturbation of this walk where vertices, or equivalently edges, that have been visited before have higher weight. We state a result of monotonicity for the speed of the once-reinforced random walk on regular trees when the bias is large and the reinforcement is small enough, see Theorem 1.12. The proof of Theorem 1.12 is inspired by work of Ben Arous, Fribergh and Sidoravicius [4], who proved a partial monotonicity result for biased random walks on Galton-Watson trees via a nice and natural coupling. We have taken some care to extract the general features of the argument (which are relatively simple) and the details of the coupling for our particular setting (which are highly non-trivial). This general method provides a coupling alternative to expansion techniques (e.g. [10]) when the self-interacting random walk has a large bias independent of the history
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