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

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 is an element of (0, 1/2] such that max(0 <= i <= n) vertical bar X-i vertical bar behaves asymptotically like n(nu). The value of. is explicitly formulated in terms of the distribution of the environment.