Resumen de The slow regime of randomly biased walks on trees
Yueyun Hu, Zhan Shi
We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk (Xn)(Xn) is null recurrent, making a maximal displacement of order of magnitude (logn)3(logn)3 in the first nn steps. We study the localization problem of XnXn and prove that the quenched law of XnXn can be approximated by a certain invariant probability depending on nn and the random environment. As a consequence, we establish that upon the survival of the system, |Xn|(logn)2|Xn|(logn)2 converges in law to some non-degenerate limit on (0,∞)(0,∞) whose law is explicitly computed.
