Convergence to equilibrium for FA-1f from any initial law with at least one infection

Establish that for the one-dimensional Fredrickson–Andersen 1-spin facilitated model (FA-1f) on the integer lattice, for any infection density q in (0,1) and any local observable f depending on finitely many sites, the expectation under any initial distribution ν that almost surely contains at least one infected site satisfies lim_{t→∞} |E_ν[f(η(t))] − π(f)| = 0, where π is the Bernoulli(q) product measure reversible for FA-1f.

Background

The FA-1f model is a prototypical kinetically constrained model (KCM) in which a site can update only if at least one of its nearest neighbors is infected. Although π, the Bernoulli(q) product measure, is a trivial equilibrium measure, proving convergence to equilibrium from non-stationary initial conditions is difficult due to non-attractiveness and constraint-induced degeneracy of rates.

The conjecture asks for global convergence to the reversible measure from any initial law that almost surely has at least one infection, for any q in (0,1). Current rigorous tools have established results only for sufficiently large q; the East model is one of the few cases where global convergence is known for all q. This problem targets a fundamental aspect of long-time dynamics out of equilibrium for FA-1f.