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 γ-stable critical Galton-Watson trees, where γ∈(1,2] and the joining x to its parent has rescaled initial weight d(O,x)α some α≤1. This corresponds recurrent regime of weights. then establish fine asymptotics limit process. In transient regime, we also give upper bound walk displacement in discrete setting, showing that never positive speed, even when weights are strongly biased away from root.