Abstract
We are interested in the biased random walk on a supercritical Galton–Watson tree in the sense of Lyons (Ann. Probab. 18:931–958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249–264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system’s non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)3, converges almost surely to a known positive constant.
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Faraud, G., Hu, Y. & Shi, Z. Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Relat. Fields 154, 621–660 (2012). https://doi.org/10.1007/s00440-011-0379-y
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DOI: https://doi.org/10.1007/s00440-011-0379-y
Keywords
- Biased random walk on a Galton–Watson tree
- Branching random walk
- Slow movement
- Random walk in a random environment
Mathematics Subject Classification (2010)
- 60J80
- 60G50
- 60K37