The Annals of Applied Probability

Continuous-time vertex reinforced jump processes on Galton–Watson trees

Anne-Laure Basdevant and Arvind Singh

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Abstract

We consider a continuous-time vertex reinforced jump process on a supercritical Galton–Watson tree. This process takes values in the set of vertices of the tree and jumps to a neighboring vertex with rate proportional to the local time at that vertex plus a constant $c$. The walk is either transient or recurrent depending on this parameter $c$. In this paper, we complete results previously obtained by Davis and Volkov [Probab. Theory Related Fields 123 (2002) 281–300, Probab. Theory Related Fields 128 (2004) 42–62] and Collevecchio [Ann. Probab. 34 (2006) 870–878, Electron. J. Probab. 14 (2009) 1936–1962] by proving that there is a unique (explicit) positive $c_{\mbox{crit}}$ such that the walk is recurrent for $c\leq c_{\mbox{crit}}$ and transient for $c>c_{\mbox{crit}}$.

Article information

Source
Ann. Appl. Probab. Volume 22, Number 4 (2012), 1728-1743.

Dates
First available in Project Euclid: 10 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1344614210

Digital Object Identifier
doi:10.1214/11-AAP811

Mathematical Reviews number (MathSciNet)
MR2985176

Zentralblatt MATH identifier
1260.60174

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

Keywords
Reinforced processes phase transition random walks on trees branching processes

Citation

Basdevant, Anne-Laure; Singh, Arvind. Continuous-time vertex reinforced jump processes on Galton–Watson trees. Ann. Appl. Probab. 22 (2012), no. 4, 1728--1743. doi:10.1214/11-AAP811. https://projecteuclid.org/euclid.aoap/1344614210


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References

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