Exponential relaxation for FA-1f under exponentially decaying infection gaps

Determine whether for the one-dimensional FA-1f model on the integer lattice, if the initial distribution ν satisfies ν(no infected vertices in [−ℓ,ℓ]) = O(e^{−κℓ}) for some κ > 0 and all sufficiently large ℓ, then for all q > 0 the convergence E_ν[f(η(t))] → π(f) is exponentially fast for every local observable f.

Background

Beyond mere convergence, quantifying the speed of relaxation to equilibrium for FA-1f is a central question. The conjecture posits exponential convergence under a natural stronger condition on the initial law—exponential tails for the event of finding no infections in large symmetric intervals.

Non-attractiveness and logarithmic Sobolev constant issues hinder standard methods. Available results establish exponential relaxation only when q is sufficiently large, often via comparison to contact processes or oriented models, underscoring the difficulty of treating small-q regimes.