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

2006/03/15 by Hu, Yueyun, Shi, Zhan
#60G50 #60K37 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.math/0603363

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 ν∈ (0, 1\over 2] such that max_0≤ i ≤ n |X_i| behaves asymptotically like n^ν. The value of ν is explicitly formulated in terms of the distribution of the environment.

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