Cutoff for the transience and mixing time of a SSEP with traps and consequences on the FEP

Abstract

We introduce a new particle system that we call the SSEP with traps, which is non reversible, attractive, and has a transient regime. We study its \emph{transience time} ${\theta }_K$, meaning the time after which the system is no longer in a transient state with high probability, on the ring with $K$ sites. We first show that ${\theta }_K$ is of order $K^2\log K$ for a system of size $K$, and more precisely that it exhibits a cutoff at time $\frac{1}{{\pi }^2}{K}^2\log K$. We then show that its \emph{mixing time} also undergoes cutoff at the same time. We further define a new mapping between the SSEP with traps and the Facilitated Exclusion Process (FEP) which has attracted significant scrutiny in recent years. We expect that this mapping will be a very useful tool to study the FEP's microscopic and macroscopic behaviour. In particular, using this mapping, we show that the FEP's transience time also undergoes a cutoff at time $\frac{1}{4{\pi }^2}{N}^2\log N$. Notably, our results show that for a FEP with particle density strictly greater than $\frac{1}{2}$, the transient component is exited in a diffusive time. This allows to extend the upper-bound from [Ayre Chleboun 2024] for the mixing time of the FEP with particle density $\rho >1/2$.