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

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.