Critical drift estimates for the frog model on trees
Emma Baileya, Matthew Jungeb, Jiaqi Liuc
- aCity University of New York, The Graduate Center
- bCity University of New York, Baruch College
- cUniversity of Pennsylvania
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Abstract
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 supd≥3pd≤1∕3. We improve this bound to 5∕17 with a shorter argument that generalizes to give bounds on supd≥mpd. We additionally prove that lim suppd≤1∕6 by finding the limiting critical drift for a non-backtracking variant.
TL;DRGuo, Tang, and Wei prove that $\limsup p_d \leq 1/6$ is the limiting critical drift for a non-backtracking variant, and improve this bound to $5/17$ with a shorter argument that generalizes to give bounds on $\sup_{d \geq m} p_D$. (Semantic Scholar)