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 [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (Xn) 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 ν ∈ (0, 12 ] 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.