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Summary. We are interested in the random walk in random environ- ment on an infinite tree. Lyons and Pemantle (11) give a precise recur- rence/transience criterion. Our paper focuses on the almost sure asymp- totic 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  ∈ (0, 12) such that max0≤i≤n |Xi| behaves asymptotically like nν. The value ofis explicitly formulated in terms of the distribution of the environment.

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