The critical-density recurrence conjecture for the frog model on non-amenable graphs

Let $G$ be a connected non-amenable graph of bounded degree, and let $\lambda$ be the frog-model particle density. Critical-density recurrence conjecture. There exists ${\lambda}_{\mathrm c}>0$ such that for all ${\lambda}>{\lambda}_{\mathrm c}$ the frog model on $G$ is ${\mathbb P}_{\lambda}$-almost surely recurrent, whereas for all ${\lambda}<\lambda_{\mathrm c}$ it is ${\mathbb P}_{\lambda}$-almost surely transient. This conjectures a positive phase-transition density, analogous to the dichotomy stated for the social-network model on infinite graphs; the source does not establish the frog-model phase transition.