Fredrickson Andersen Model and Noisy Majority Vote Process on Nonamenable Graphs

Abstract: We study the Fredrickson-Andersen j-spin facilitated model and the noisy majority vote process on connected infinite graphs satisfying suitable expansion properties. For the former, we consider the out-of-equilibrium regime where the density of facilitating sites is close to 1, both for the equilibrium product measure and for the initial configuration, and we show exponential convergence to equilibrium. For the latter, we prove the existence of multiple equilibrium measures, generalising recent results by J. Ding and F. Huang (2025). Our proofs build on the framework of decorated set systems introduced by I. Hartarsky and F. Toninelli (2024) and establish exponentially decaying tails for the diameter of the space-time cluster of zeros containing a fixed vertex for both perturbed bootstrap percolation and consensus processes. The results are essentially sharp on hyperbolic lattices and we further show how some of the borderline cases can be handled using Toom contours, in the reformulation by J. M. Swart, R. Szab\'o and C. Toninelli (2022).