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

Yueyun Hu (1) , Zhan Shi (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: Random walk; random environment; tree; Mandelbrot's multiplicative; cascade

Domaines: Probabilités [math.PR]

Yueyun Hu, Zhan Shi. A subdiffusive behaviour of recurrent random walk in random environment on a regular tree. 2006. ⟨hal-00020751⟩