Mixing Times for the Facilitated Exclusion Process

Abstract

The facilitated simple exclusion process (FEP) is a one-dimensional exclusion process with a dynamical constraint. We establish bounds on the mixing time of the FEP on the segment, with closed boundaries, and the circle. The FEP on these spaces exhibits transient states that, if the macroscopic density of particles is at least $1/2$, the process will eventually exit to reach an ergodic component. If the macroscopic density is less than $1/2$ the process will hit an absorbing state. We show that the symmetric FEP (SFEP) on the segment $\{1,\ldots,N\}$, with $k>N/2$ particles, has mixing time of order $N^{2}\log(N-k)$ and exhibits the pre-cutoff phenomenon. For the asymmetric FEP (AFEP) on the segment, we show that there exists initial conditions for which the hitting time of the ergodic component is exponentially slow in the number of holes $N-k$. In particular, when $N-k$ is large enough, the hitting time of the ergodic component determines the mixing time. For the SFEP on the circle of size $N$, and macroscopic particle density $ρ\in(1/2,1)$, we establish bounds on the mixing time of order $N^{2}\log N$ for the process restricted to its ergodic component. We also give an upper bound on the hitting time of the ergodic component of order $N^{2}\log N$ for a large class of initial conditions. The proofs rely on couplings with exclusion processes (both open and closed boundaries) via a novel lattice path (height function) construction of the FEP.

Figures (9)

• Figure 1: Transition rates for the FEP on the segment. The indicated jumps are the only ones that may occur in the dynamics.
• Figure 2: An ergodic configuration in $\mathcal{E}_{8,6}$.
• Figure 3: Transition rates for the SFEP on the circle. The indicated jumps are the only ones which may occur in the dynamics. The configuration in $\Omega_{15,9}^{\circ}$ does not belong to the ergodic component $\mathcal{G}_{N,k}$ as there are two adjacent holes.
• Figure 4: (Ergodic regions) The figure above represents a configuration in $\mathcal{G}_{20,11}^{c}\subset\Omega_{20,11}^{\circ}$. Black circles denote particles and white circles denote holes. The The ergodic regions from Definition \ref{['def:regions']} are given by the intervals $I_{1}=[14,19],I_{2}=[2,6]$ and $I_{3}=[9,11]$.
• Figure 5: In the above figure black circles denote particles and white circles denote holes. Particle--hole objects are placed in red boxes. In the dynamics, the third leftmost particle in the top figure jumps left, which corresponds to the leftmost particle--hole object jumping one space to the right. In particular, the boxes do not overlap in the dynamics and particle--hole objects behave like particles in the simple exclusion process.

Theorems & Definitions (34)

• Definition 2.1
• Definition 2.2
• Theorem 2.1: SFEP
• Theorem 2.2: AFEP
• Definition 2.3
• Theorem 2.3
• Remark 2.4
• Remark 3.1
• Proposition 3.1
• proof : Proof of Proposition \ref{['prop:partial']}