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Cutoff for the mixing time of the Facilitated Exclusion Process

Brune Massoulié

TL;DR

This work analyzes the mixing time of the Facilitated Exclusion Process (FEP) on the discrete circle in the supercritical regime ($K>N/2$), where transient and ergodic states coexist. A central contribution is a bijective static/dynamic mapping from an ergodic FEP together with its current to a Symmetric Simple Exclusion Process (SSEP) with current through the origin, augmented by a height-function representation and the corner-flip dynamics. By decomposing the approach into transience and ergodic mixing, and leveraging sharp SSEP techniques, the authors establish a cutoff for the worst-case mixing time at scale $\tau_N(\varepsilon) \sim \frac{1}{4\pi^2} N^2 \log N$ as $N\to\infty$, with regime-dependent refinements as the particle count $K$ varies near the critical threshold. The bijection also yields a deterministic coupling between a FEP tagged particle position and an SSEP-origin current, and opens avenues for extending the methodology to broader settings and sharper cutoff windows.

Abstract

We compute the mixing time of the Facilitated Exclusion Process (FEP) and obtain cutoff and pre-cutoff in different regimes. The main tool to obtain this result is a new bijective, deterministic mapping between the joint law of an ergodic FEP and its current through the origin, and the joint law of a Symmetric Simple Exclusion Process (SSEP) and its current through the origin. This mapping is interesting in itself, as it remains valid in the non-ergodic regime where it gives a coupling between the position of a tagged particle in the FEP and the current through the origin in a SSEP with traps.

Cutoff for the mixing time of the Facilitated Exclusion Process

TL;DR

This work analyzes the mixing time of the Facilitated Exclusion Process (FEP) on the discrete circle in the supercritical regime (), where transient and ergodic states coexist. A central contribution is a bijective static/dynamic mapping from an ergodic FEP together with its current to a Symmetric Simple Exclusion Process (SSEP) with current through the origin, augmented by a height-function representation and the corner-flip dynamics. By decomposing the approach into transience and ergodic mixing, and leveraging sharp SSEP techniques, the authors establish a cutoff for the worst-case mixing time at scale as , with regime-dependent refinements as the particle count varies near the critical threshold. The bijection also yields a deterministic coupling between a FEP tagged particle position and an SSEP-origin current, and opens avenues for extending the methodology to broader settings and sharper cutoff windows.

Abstract

We compute the mixing time of the Facilitated Exclusion Process (FEP) and obtain cutoff and pre-cutoff in different regimes. The main tool to obtain this result is a new bijective, deterministic mapping between the joint law of an ergodic FEP and its current through the origin, and the joint law of a Symmetric Simple Exclusion Process (SSEP) and its current through the origin. This mapping is interesting in itself, as it remains valid in the non-ergodic regime where it gives a coupling between the position of a tagged particle in the FEP and the current through the origin in a SSEP with traps.

Paper Structure

This paper contains 17 sections, 26 theorems, 117 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Structure of the proof
  4. A crucial mapping
  5. Static mapping
  6. Dynamic mapping
  7. Proof of the upper bound
  8. The height function representation
  9. Coupling with near-equilibrium
  10. Distribution of the initial height
  11. Upper bound when the number of particles is close to the edges
  12. Initial height
  13. Time to couple the height functions
  14. Proof of the lower bound
  15. Properties of the eigenfunctions
  16. ...and 2 more sections

Key Result

Theorem 2.1

For all $\varepsilon \in (0,1)$, there exists $C_\varepsilon >0$ such that for all $N$,

Figures (4)

  • Figure 1: Illustration of the static mapping. The rank of the purple particle in $\eta$ is $k$ and its position is $X_k$. Each site of $\sigma$ is in correspondence with a particle of $\eta$: the first site of $\sigma$ is related to the $k^{th}$ particle of $\eta$, the second site of $\sigma$ to the $(k+1)^{th}$ particle of $\eta$, etc. If a particle of $\eta$ is followed by another particle, it is underlined in blue in $\eta$, and there is a particle on the corresponding site of $\sigma$. If a particle of $\eta$ is followed by an empty site, it is underlined in orange in $\eta$, and the corresponding site of $\sigma$ is empty.
  • Figure 2: Different kinds of jumps. The tagged particle is coloured in purple.
  • Figure 3: Illustration of the dynamic mapping and the link between current and initial height.
  • Figure 4: Summary of the coupling strategy. The maximum height difference of $\zeta$ is controlled by Proposition \ref{['prop:flu_ssep']}. This gives us the inequality $\zeta^{(2)} \le \zeta \le \zeta^{(1)}$. Lemma \ref{['lem:init_height']} tells us the initial heights of $\zeta^{(1)}$ and $\zeta^{(2)}$ modulo $N$ are close to being uniform. So when all height functions are coupled, $(X(t), \sigma(t))$ is close to $\mathcal{U}(\mathbb{T}_N) \otimes \pi_{K,P}^{\textsc{ssep}}$.

Theorems & Definitions (51)

  • Definition 2.1: Cutoff and pre-cutoff
  • Theorem 2.1: $K$-uniform cutoff for the mixing time
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2: Cutoff and pre-cutoff as a function of $K$
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2: Transience time estimate
  • Proposition 3.3: Upper bound, far from the edges
  • ...and 41 more