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 is an element of (1, 2] and where the edge joining x to its parent has rescaled initial weight d(O, x)alpha for some alpha <= 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.

Automated summary

An invariance principle is proven for linearly edge reinforced random walks on gamma stable critical Galton-Watson trees, with gamma in (1,2] and initial edge weight rescaled by d(O,x)^alpha, for alpha<=1. This corresponds to the recurrent regime of initial weights. Fine asymptotics for the limit process are established. In the transient regime, an upper bound on the random walk displacement is given, demonstrating that the edge reinforced random walk never has positive speed, even with biased initial edge weights away from the root.