We prove an invariance principle for linearly edge reinforced random walks on γ-stable critical Galton-Watson trees, where and where the edge joining x to its parent has rescaled initial weight for some . 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.
The research of EA was supported by JSPS and ERC starting grant 676970 RANDGEOM.
We would like to thank David Croydon for introducing us to this topic, and Andrea Collevecchio for useful comments on the manuscript.
"Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees." Electron. J. Probab. 28 1 - 64, 2023. https://doi.org/10.1214/23-EJP901