The slow regime of randomly biased walks on trees

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) is null recurrent, making a maximal displacement of order of magnitude (logn)3 in the first n steps. We study the localization problem of Xn and prove that the quenched law of Xn can be approximated by a certain invariant probability depending on n and the random environment. As a consequence, we establish that upon the survival of the system, |Xn|(logn)2 converges in law to some non-degenerate limit on (0,.) whose law is explicitly computed.