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 (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 ν ∈ (0,1/2] such that max0≤i≤n|Xi| behaves asymptotically like nν. The value of ν is explicitly formulated in terms of the distribution of the environment. © Springer-Verlag 2007.