The slow regime of randomly biased walks on trees

2015/01/30 by Yueyun Hu, Hu, Yueyun, Zhan Shi +1
Mathematics · #60G50 #60J80 #60K37 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G50 #msc:60J80 #msc:60K37

paper · pdf · doi:10.48550/arxiv.1501.07700

43 pages. We added a recent work by Jim Pitman ([38]) for the limiting law

arxiv created 2015/09/26 · arxiv updated 2015/09/29

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, (|X_n|)/((log n)²) converges in law to some non-degenerate limit on (0, ∞) whose law is explicitly computed.

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