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

Abstract

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] 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 [13]'s multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent $\nu \in (0,\frac{1}{2}]$ such that $\max \_0\le i\le n|X\_i|$ behaves asymptotically like $n^{\nu }$. The value of $\nu$ is explicitly formulated in terms of the distribution of the environment.

29 pages with 1 figure. Its preliminary version was put in the following web site: http://www.math.univ-paris13.fr/prepub/pp2005/pp2005-28.html