Scaling limit of the recurrent biased random walk on a Galton-Watson tree

TL;DR

This paper proves that the scaled trace of a null recurrent biased random walk on a Galton-Watson tree converges to the Brownian forest, extending to random environments.

Contribution

It establishes the scaling limit of biased random walks on Galton-Watson trees, including in random environments, as a convergence to the Brownian forest.

Findings

Trace converges to Brownian forest after proper renormalization

Results extend to random walk in random environment

Provides a new understanding of scaling limits in random trees

Abstract

We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.