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November 2016 The slow regime of randomly biased walks on trees
Yueyun Hu, Zhan Shi
Ann. Probab. 44(6): 3893-3933 (November 2016). DOI: 10.1214/15-AOP1064

Abstract

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.

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Yueyun Hu. Zhan Shi. "The slow regime of randomly biased walks on trees." Ann. Probab. 44 (6) 3893 - 3933, November 2016. https://doi.org/10.1214/15-AOP1064

Information

Received: 1 October 2014; Revised: 1 September 2015; Published: November 2016
First available in Project Euclid: 14 November 2016

zbMATH: 1360.60156
MathSciNet: MR3572327
Digital Object Identifier: 10.1214/15-AOP1064

Subjects:
Primary: 60G50, 60J80, 60K37

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 6 • November 2016
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