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} $θ_K$, 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 $θ_K$ is of order $K^2 \log K$ for a system of size $K$, and more precisely that it exhibits a cutoff at time $\frac{1}{π^2} K^2 \log K$. 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 $\frac{1}{4 π^2} N^2 \log N$. Notably, our results show that for a FEP with particle density strictly greater than $\frac12$, 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 $ρ> 1/2$.

Figures (14)

• Figure 1: A local configuration of the SWT, with two jumps represented: the jump of the blue particle to an empty neighbour, and the jump of the red particle to a neighbouring trap, killing it and reducing the trap's depth by $1$.
• Figure 2: Examples of the different types of configuration of the FEP on $\mathbb{T}_{11}$
• Figure 3: Illustration of the statical mapping $\eta\mapsto \xi$, which associates particles in $\eta$ with sites in $\xi$. Particles that are followed by another particle in the $\eta$ are represented in blue and become occupied sites in $\xi$. Particles that are followed by one or more empty sites in $\eta$ are represented in red and become empty sites or traps in $\xi$.
• Figure 4: When a FEP configuration $\eta$ is ergodic, the mapped configuration $\xi=\Pi(\eta)$ is a SSEP configuration.
• Figure 5: When a FEP configuration $\eta$ is frozen, the mapped configuration $\xi=\Pi(\eta)$ only has traps, no particles.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2: Cutoff for the transience time of the SSEP with traps
• Remark 1: Worst configurations and optimal cutoff window
• Remark 2: Transience time for independent random walks
• Theorem 1.3: Cutoff for the mixing time of the SWT
• Remark 3
• Remark 4: Mixing time for independent random walks
• Remark 5: No negative dependence for the SWT
• Remark 6: Attractiveness and monotonicity of the transience probability
• Theorem 2.1