Scaling exponents of random walks in random sceneries

Abstract

We compute the exact asymptotic normalizations of random walks in random sceneries, for various null recurrent random walks to the nearest neighbours, and for i.i.d., centered and square integrable random sceneries. In each case, the standard deviation grows like n α with α∈[ 1 2 ;1] . Here, the value of the exponent α is determined by the sole geometry of the underlying graph, as opposed to previous examples, where this value reflected mainly the integrability properties of the steps of the walk, or of the scenery. For discrete Bessel processes of dimension d∈[0;2[, the exponent is α= max{1−d/2, 3 4 } . For the simple walk on some specific graphs, whose volume grows like n d for d∈[1;2[, the exponent is α=1− d/4. We build a null recurrent walk, for which α= 1 2 without logarithmic correction. Last, for the simple walk on a critical Galton–Watson tree, conditioned by its nonextinction, the annealed exponent is α= 2 3 . In that setting and when the scenery is i.i.d. by levels, the same result holds with α= 5 6 .