# Invariance principles for random walks in random environment on trees

## TL;DR

This paper proves invariance principles for random walks in random environments on trees, demonstrating convergence to Brownian motion in Gaussian potentials on continuum random trees, with applications to various branching structures.

## Contribution

It establishes new functional limit theorems for biased and edge-reinforced random walks on trees, extending invariance principles to complex random environments.

## Key findings

- Biased random walk converges to Brownian motion in Gaussian potential on CRT.
- Edge-reinforced random walk scales to Brownian motion with drift on CRT.
- Results apply to critical Galton-Watson trees and non-lattice branching random walks.

## Abstract

In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees. First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on $\mathbb{R}^d$ indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in $\mathbb{R}^d$. This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the recent common ancestor. We prove, after introducing the bias parameter $\beta^{n^{-1/4}}$, for some $\beta>1$, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous' continuum random tree (CRT). Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.