Open Access
2024 Critical drift estimates for the frog model on trees
Emma Bailey, Matthew Junge, Jiaqi Liu
Author Affiliations +
Electron. J. Probab. 29: 1-21 (2024). DOI: 10.1214/24-EJP1108


Place an active particle at the root of a d-ary tree and a single dormant particle at each non-root site. In discrete time, active particles move towards the root with probability p and, otherwise, away from the root to a uniformly sampled child vertex. When an active particle moves to a site containing a dormant particle, the dormant particle becomes active. The critical drift pd is the infimum over all p for which infinitely many particles visit the root almost surely. Guo, Tang, and Wei proved that supd3pd13. We improve this bound to 517 with a shorter argument that generalizes to give bounds on supdmpd. We additionally prove that lim suppd16 by finding the limiting critical drift for a non-backtracking variant.

Funding Statement

All authors were partially supported by NSF grant #2115936. Junge was partially supported by NSF grant #2238272.


We are grateful to the authors of [9] whose code formed the basis of the simulations performed in Section 6. We would also like to thank Serguei Popov for sending us an electronic copy of [3] whose result is applied in the proof of Lemma 4.2.


Download Citation

Emma Bailey. Matthew Junge. Jiaqi Liu. "Critical drift estimates for the frog model on trees." Electron. J. Probab. 29 1 - 21, 2024.


Received: 18 May 2023; Accepted: 1 March 2024; Published: 2024
First available in Project Euclid: 27 March 2024

Digital Object Identifier: 10.1214/24-EJP1108

Primary: 60K35

Keywords: Interacting particle system , phase transition

Vol.29 • 2024
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