The number of generations entirely visited for recurrent random walks on random environment

TL;DR

This paper investigates the behavior of recurrent random walks on super-critical Galton-Watson trees in random environments, showing that the largest fully visited generation scales logarithmically with the number of steps, with a constant linked to branching random walk properties.

Contribution

It establishes the asymptotic behavior of the largest visited generation in recurrent random walks on Galton-Watson trees, connecting it to Biggins' law of large numbers.

Findings

Largest visited generation scales as log n

Normalization constant relates to Biggins' law

Behavior extends previous recurrence studies

Abstract

In this paper we are interested in a random walk in a random environment on a super-critical Galton-Watson tree. We focus on the recurrent cases already studied by Y. Hu and Z. Shi and G. Faraud. We prove that the largest generation entirely visited by these walks behaves like log n and that the constant of normalization which differs from a case to another is function of the inverse of the constant of Biggins' law of large number for branching random walks.