Sharp freezing time estimates for the subcritical Facilitated Exclusion Process

We investigate the exact transience time of the Facilitated Exclusion Process (FEP) on the one-dimensional torus with N sites. The FEP exhibits an active/inactive phase transition at critical density $1/2$, such that in the subcritical density regime $(0,1/2)$, it becomes frozen after a finite time period -- the transience time or freezing time. We first show that for the FEP starting from a Bernoulli product measure of marginal density ρ∈ (0,1/2), the transience time has exactly the scale of Θ(log³ N). Secondly, we prove that in the near-critical case ρ 1/2 - N-α for α∈ (0,1), the transience time is polynomial and has a scale of N1 (2α). The key idea is to estimate the typical size of locally supercritical intervals of the initial distribution, which has order log N in the subcritical case and N1 (2α) in the near-critical case. In the subcritical case this is enough, whereas in the near-critical case we need additional dynamical decorrelation inequalities to apply this static result to estimate the freezing time.