# A quenched central limit theorem for biased random walks on supercritical Galton-Watson trees

## TL;DR

This paper proves a quenched functional central limit theorem for biased random walks on supercritical Galton-Watson trees with leaves, extending previous results and confirming a sharp bias bound conjectured by others.

## Contribution

It extends the quenched CLT to trees with leaves and verifies the sharpness of the bias bound conjectured by Ben Arous and Fribergh.

## Key findings

- Established a quenched functional CLT for biased walks on trees with leaves.
- Confirmed the sharpness of the bias upper bound conjectured earlier.
- Extended previous results to more general Galton-Watson trees.

## Abstract

In this note, we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton-Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves is considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.