Improved critical drift estimates for the frog model on trees

TL;DR

This paper improves bounds on the critical drift probability in the frog model on infinite trees, providing new insights into the conditions for infinite activation of particles.

Contribution

It offers improved bounds on the critical drift and proves monotonicity for a self-similar variant of the frog model on trees.

Findings

Established tighter bounds on the critical drift probability $p_d$ for large $d$.

Proved the monotonicity of critical values in a self-similar frog model.

Enhanced understanding of phase transition behavior in the frog model.

Abstract

Place an active particle at the root of the infinite $d$-ary tree and dormant particles at each non-root site. Active particles move towards the root with probability $p$ and otherwise move to a uniformly sampled child vertex. When an active particle moves to a site containing dormant particles, all the particles at the site become active. The critical drift $p_d$ is the infimum over all $p$ for which infinitely many particles visit the root almost surely. We give improved bounds on $\sup_{d \geq m} p_d$ and prove monotonicity of critical values associated to a self-similar variant.