The slow regime of randomly biased walks on trees

• Yueyun Hu ^[2] ; Zhan Shi ^[1]
  1. [1] Pierre and Marie Curie University

     Pierre and Marie Curie University

     París, Francia

     
  2. [2] Université Paris XIII
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 6, 2016, págs. 3893-3933
• Idioma: inglés
• DOI: 10.1214/15-aop1064
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• Resumen

  • 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 (Xn)(Xn) is null recurrent, making a maximal displacement of order of magnitude (logn)3(log⁡n)3 in the first nn steps. We study the localization problem of XnXn and prove that the quenched law of XnXn can be approximated by a certain invariant probability depending on nn and the random environment. As a consequence, we establish that upon the survival of the system, |Xn|(logn)2|Xn|(log⁡n)2 converges in law to some non-degenerate limit on (0,∞)(0,∞) whose law is explicitly computed.