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Mixing time for the asymmetric simple exclusion process in a random environment

Hubert Lacoin, Shangjie Yang

Abstract

We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\in [ 1, N]$ jumps to site $x-1$ (if $x\ge 2$) at rate $1-ω_x$ and to site $x+1$ (if $x \le N-1$) at rate $ω_x$ if the target site is not occupied. The sequence $ω=(ω_x)_{ x \in \mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\mathbb{E}[ \log ρ_1 ]<0$ where $ρ_1:= (1-ω_1)/ω_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{β+o(1)}$ particles where $β\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\max\left(1,\frac {1}λ, β+ \frac 1 {2λ}\right)+o(1)} \le t_{\mathrm{Mix}}^{N,k} \le N^{C+o(1)}$$ where $λ>0$ is such that $\mathbb{E}[ρ_1^λ]=1$ ($λ=\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $ω$. We conjecture that our lower bound is sharp up to sub-polynomial correction.

Mixing time for the asymmetric simple exclusion process in a random environment

Abstract

We consider the simple exclusion process in the integer segment with particles and spatially inhomogenous jumping rates. A particle at site jumps to site (if ) at rate and to site (if ) at rate if the target site is not occupied. The sequence is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume where , which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of . More precisely, for the exclusion process with particles where , we have in the large asymptotic where is such that ( if the equation has no positive root) and is a constant which depends on the distribution of . We conjecture that our lower bound is sharp up to sub-polynomial correction.

Paper Structure

This paper contains 36 sections, 24 theorems, 224 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. The exclusion process in a random environment
  4. Model and result
  5. An introduction to Random Walk in a Random Environment $\omega$
  6. The Simple Exclusion process in an environment $\omega$
  7. Definition
  8. Mixing time and spectral gap
  9. Results
  10. Universal bounds for the mixing time on the exclusion process
  11. Mixing time for the exclusion process in a random environment
  12. Related work
  13. Mixing time for the exclusion process in a homogeneous environment
  14. Mixing time for the random walk in a random environment
  15. Mixing time for the exclusion in a ballistic environment
  16. ...and 21 more sections

Key Result

Proposition 2.1

For any $k\in \llbracket 1,N/2\rrbracket$ and $N\ge 2$, for any $(\omega_x)_{x\in {\mathbb Z} }$ we have Furthermore, if $k_N$ is a sequence such that we have for any $\varepsilon> 0$, for $N\ge N_0(\varepsilon)$ sufficiently large for any $(\omega_x)_{x\in {\mathbb Z} }$

Figures (5)

  • Figure 1: A graphical representation of the simple exclusion process in the segment $\llbracket 1, N \rrbracket$ and environment $\omega=(\omega_x)_{x \in {\mathbb Z} }$: a bold circle represents a particle, and the number above every arrow represents the jump rate while a red $"\times"$ represents an inadmissible jump.
  • Figure 2: A graphical description of the function $F(u)$ with only two zeros at $u=0$ and $u=\lambda$.
  • Figure 3: The phase diagram for the exponent of the mixing time (the lower bound is proved rigorously and the upper bound is only conjectured). The transition between the blue and red (hatched) regions of the diagram corresponds to the transition of the RWRE from the ballistic phase to the transient-with-zero-speed phase. A third phase represented by the white region appears when one considers a large number of particles, in this phase the main limitation to mixing is the flow of particle through the deepest trap.
  • Figure 4: A bold circle represents a particle, and a particle at the same site for the configurations $\xi^{(i-1)}$ and $\xi^{(i)}$ is colored black. Otherwise, it is red or blue. (L) A graphical description of the movements of the particle at site $x_i$ of $\xi^{(i-1)}$ to the empty site $y_i$ and the numbers above the arrows are the relative order of the movements. (R) We draw the graph of $(\ell, V(\xi_{\ell}^{(i)}))_{\ell}$.
  • Figure 5: A graphical representation of the boundary driven process: a bold circle represents a particle, and the number above every arrow represents the jump rate while a red $"\times"$ represents an inadmissible jump. In addition, the site $y_2+1$ can accommodate infinite many particles and all particles at site $y_2+1$ stay put.

Theorems & Definitions (43)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 3.1
  • Proposition 3.2
  • proof : Sketch of proof
  • Proposition 3.3
  • proof
  • ...and 33 more