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

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

2006-03-15

arxiv.org/abs/math/0603363v1

Mathematics

Probability

29 pages with 1 figure. Its preliminary version was put in the following web site: http://www.math.univ-paris13.fr/prepub/pp

Scientific paper

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 $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$ behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly formulated in terms of the distribution of the environment.

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