A subdiffusive behaviour of recurrent random walk in random environment on a regular tree. (English) Zbl 1113.60096
Summary: We are interested in the random walk in random environment on an infinite tree. R. Lyons and R. Pemantle [Ann. Probab. 20, No. 1, 125–136 (1992; Zbl 0751.60066)] 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 B. B. Mandelbrot’s multiplicative cascade [C. R. Acad. Sci., Paris, Sér. A 278, 289–292 (1974; Zbl 0276.60096)]. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent \(\nu\in (0, \frac 12 ]\) such that \(\max_{0\leq i \leq n} |X_i|\) behaves asymptotically like \({n^{\nu}}\). The value of \(\nu\) is explicitly formulated in terms of the distribution of the environment.
MSC:
| 60K37 | Processes in random environments |
| 60G50 | Sums of independent random variables; random walks |
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