The heavy range of randomly biased walks on trees

Abstract

We focus on recurrent random walks in random environment (RWRE) on Galton–Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers Andreoletti and Chen (2018), Aïdékon and de Raphélis (2017) and de Raphélis (2016). Here we study the heavy range: the number of edges frequently visited by the walk. The asymptotic behavior of this process when the number of visits is a power of the number of steps of the walk is given for all recurrent cases. It turns out that this heavy range plays a crucial role in the rate of convergence of an estimator of the environment from a single trajectory of the RWRE.