The slow regime of randomly biased walks on trees

Abstract

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 $(X_n)$ is null recurrent, making a maximal displacement of order of magnitude $(\log n{)}^3$ in the first $n$ steps. We study the localization problem of $X_n$ and prove that the quenched law of $X_n$ 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, $\frac{|X_n|}{(\log n{)}^2}$ converges in law to some non-degenerate limit on $(0,\infty )$ whose law is explicitly computed.

43 pages. We added a recent work by Jim Pitman ([38]) for the limiting law

References in corpus (4)

• The maximal potential energy of biased random walks on trees
• The maximal drawdown of the Brownian meander
• First order transition for the branching random walk at the critical parameter
• The most visited sites of biased random walks on trees

Cited by in corpus (2)

• The maximal drawdown of the Brownian meander
• The most visited sites of biased random walks on trees