Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees

TL;DR

This paper establishes an invariance principle and asymptotic behavior for linearly edge-reinforced random walks on critical Galton-Watson trees, revealing recurrence and displacement bounds in different regimes.

Contribution

It introduces a new invariance principle for edge-reinforced random walks on critical Galton-Watson trees with detailed asymptotics and displacement bounds.

Findings

Invariance principle for recurrent regime on critical trees

Fine asymptotics for the limit process

Upper bounds on displacement showing zero speed in transient regime

Abstract

We prove an invariance principle for linearly edge reinforced random walks on $\gamma$-stable critical Galton-Watson trees, where $\gamma \in (1,2]$ and where the edge joining $x$ to its parent has rescaled initial weight $d(\rho, x)^{\alpha}$ for some $\alpha \leq 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.