Open Access
2023 Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees
George Andriopoulos, Eleanor Archer
Author Affiliations +
Electron. J. Probab. 28: 1-64 (2023). DOI: 10.1214/23-EJP901


We prove an invariance principle for linearly edge reinforced random walks on γ-stable critical Galton-Watson trees, where γ(1,2] and where the edge joining x to its parent has rescaled initial weight d(O,x)α for some α1. 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.

Funding Statement

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.


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George Andriopoulos. Eleanor Archer. "Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees." Electron. J. Probab. 28 1 - 64, 2023.


Received: 7 January 2022; Accepted: 2 January 2023; Published: 2023
First available in Project Euclid: 13 January 2023

MathSciNet: MR4533740
zbMATH: 1514.60054
Digital Object Identifier: 10.1214/23-EJP901

Primary: 60F17 , 60J60 , 60K37 , 60K50

Keywords: Diffusion in random environment , Dirichlet distribution , Galton–Watson trees , Random walk in random environment , reinforced random walks , slow movement

Vol.28 • 2023
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