The Annals of Probability

On the transience of processes defined on Galton–Watson trees

Andrea Collevecchio

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We introduce a simple technique for proving the transience of certain processes defined on the random tree $\mathcal{G}$ generated by a supercritical branching process. We prove the transience for once-reinforced random walks on $\mathcal{G}$, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567–592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on $\mathcal{G}$. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b≥4 and recurrent if b=1. The case b=2 is still open.

Article information

Ann. Probab. Volume 34, Number 3 (2006), 870-878.

First available in Project Euclid: 27 June 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J75: Jump processes

Reinforced random walk random walk on trees branching processes


Collevecchio, Andrea. On the transience of processes defined on Galton–Watson trees. The Annals of Probability 34 (2006), no. 3, 870--878. doi:10.1214/009117905000000837.

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