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A subdiffusive behaviour of recurrent random walk in random environment on a regular tree

Authors:	Yueyun Hu Zhan Shi

Institution:	(1) Département de Mathématiques, Université Paris XIII, 99 avenue J-B Clément, 93430 Villetaneuse, France;(2) Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France

Abstract:	We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle (Ann. Probab. 20, 125–136, 1992) give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (X _n) in random environment on a regular tree, which is closely related to Mandelbrot’s (C. R. Acad. Sci. Paris 278, 289–292, 1974) multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent ${\nu\in \big(0, {1\over 2}\big]}$ such that ${\max_{0\le i \le n} \|X_i\|}$ behaves asymptotically like ${n^{\nu}}$ . The value of ν is explicitly formulated in terms of the distribution of the environment.

Keywords:	Random walk Random environment Tree Mandelbrot’ s multiplicative cascade
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