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Dynamics of the leftmost particle in heterogeneous semi-infinite exclusion systems

Mikhail Menshikov, Serguei Popov, Andrew Wade

TL;DR

The work analyzes the leftmost particle in a heterogeneous semi-infinite exclusion process by linking its dynamics to a customer random walk that traverses an infinite Jackson-type network of inter-particle gaps. Using M/G/∞ queue comparisons, stationary-product measures for the gap process, and Lamperti-type perturbations of the jump rates, it classifies regimes where the leftmost particle is ballistically left, positively recurrent, null recurrent, or sub-ballistically transient, and shows that the initial configuration crucially affects the outcome. It also demonstrates null recurrence via a carefully constructed rate sequence and establishes dynamic recurrence for stationary starts, highlighting rich interplay between the microscopic rate structure and macroscopic transport. The results illuminate how inhomogeneity and initial data shape drift and fluctuations of the leftmost particle, with potential implications for transport in inhomogeneous interacting particle systems.

Abstract

We study the behaviour of the leftmost particle in a semi-infinite particle system on $\mathbb{Z}$, where each particle performs a continuous-time nearest-neighbour random walk, with particle-specific jump rates, subject to the exclusion interaction (i.e., no more than one particle per site). We give conditions, in terms of the jump rates on the system, under which the leftmost particle is recurrent or transient, and develop tools to study its rate of escape in the transient case, including by comparison with an $M/G/\infty$ queue. In particular we show examples in which the leftmost particle can be null recurrent, positive recurrent, ballistically transient, or subdiffusively transient. Finally we indicate the role of the initial condition in determining the dynamics, and show, for example, that sub-ballistic transience can occur started from close-packed initial configurations but not from stationary initial conditions.

Dynamics of the leftmost particle in heterogeneous semi-infinite exclusion systems

TL;DR

The work analyzes the leftmost particle in a heterogeneous semi-infinite exclusion process by linking its dynamics to a customer random walk that traverses an infinite Jackson-type network of inter-particle gaps. Using M/G/∞ queue comparisons, stationary-product measures for the gap process, and Lamperti-type perturbations of the jump rates, it classifies regimes where the leftmost particle is ballistically left, positively recurrent, null recurrent, or sub-ballistically transient, and shows that the initial configuration crucially affects the outcome. It also demonstrates null recurrence via a carefully constructed rate sequence and establishes dynamic recurrence for stationary starts, highlighting rich interplay between the microscopic rate structure and macroscopic transport. The results illuminate how inhomogeneity and initial data shape drift and fluctuations of the leftmost particle, with potential implications for transport in inhomogeneous interacting particle systems.

Abstract

We study the behaviour of the leftmost particle in a semi-infinite particle system on , where each particle performs a continuous-time nearest-neighbour random walk, with particle-specific jump rates, subject to the exclusion interaction (i.e., no more than one particle per site). We give conditions, in terms of the jump rates on the system, under which the leftmost particle is recurrent or transient, and develop tools to study its rate of escape in the transient case, including by comparison with an queue. In particular we show examples in which the leftmost particle can be null recurrent, positive recurrent, ballistically transient, or subdiffusively transient. Finally we indicate the role of the initial condition in determining the dynamics, and show, for example, that sub-ballistic transience can occur started from close-packed initial configurations but not from stationary initial conditions.
Paper Structure (10 sections, 9 theorems, 57 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dynamics of the leftmost particle
  3. Introducing the customer random walk
  4. Finite initial configurations
  5. Overview
  6. Lower bound: comparison with the M/G/infinity queue
  7. Upper bound: using the stationary distribution
  8. Lamperti-type rates and proof of Theorem \ref{['thm:lamperti']}
  9. An example with a null-recurrent leftmost particle
  10. Stationary initial configurations: dynamic recurrence

Key Result

Proposition 1.1

Suppose that Condition ass:rates holds, and that there exists at least one admissible solution $\rho$ to eq:stable-traffic. Take $X(0) \in {\mathbb X}_{\mathrm{F}}$ with $X_1 (0) =0$. Then, with $v_0$ given by eq:v0-def, For $\rho = \rho(v_0)$ the minimal solution to eq:stable-traffic, we have that, for every finite $A \subset {\mathbb N}$, Moreover, if $v_0 <0$ then the minimal solution is the

Figures (1)

  • Figure 1: The events $E^{(1)}(t_1)$ and $E^{(2)}(t_2)$ (with $t_1<t_2$).

Theorems & Definitions (22)

  • Proposition 1.1
  • Theorem 1.3
  • Definition 1.6: Customer random walk
  • Remark 1.7
  • Remark 1.8
  • proof : Proof of Proposition \ref{['prop:known-finite']}
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4
  • ...and 12 more