Exponential relaxation of FA-1f from exponentially nonempty initial laws

Work on the one-dimensional lattice $\mathbb{Z}$ with the FA-1f model, and let $\pi$ be its Bernoulli product equilibrium measure. Let $\nu$ be an initial distribution, and consider local observables $f$ as in the preceding convergence conjecture.

Exponential-relaxation conjecture. If there exists $\kappa>0$ such that

$$\nu(\text{no infected vertices in }[-\ell,\ell])=O(e^{-\kappa\ell})$$

for all sufficiently large $\ell>0$, then for every $q>0$ the convergence to equilibrium in the preceding formula is exponentially fast.

This conjecture asks for a quantitative strengthening of convergence under an exponential-tail assumption on the size of the initially infection-free interval around the origin. The paper introduces it as the next open question after the qualitative convergence conjecture.