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Stabilization time of finite configurations with a second class particle in discrete TASEP

Bori Anna Mészáros, Bálint Vető

Abstract

We consider finite configurations of particles and holes sampled according to Bernoulli product measure and with a second class particle added to a random position. The stabilization time is the number of steps needed to reach an ordered state under discrete time TASEP dynamics with parallel update. We describe the additional time of stabilization caused by the presence of the second class particle.

Stabilization time of finite configurations with a second class particle in discrete TASEP

Abstract

We consider finite configurations of particles and holes sampled according to Bernoulli product measure and with a second class particle added to a random position. The stabilization time is the number of steps needed to reach an ordered state under discrete time TASEP dynamics with parallel update. We describe the additional time of stabilization caused by the presence of the second class particle.

Paper Structure

This paper contains 8 sections, 7 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stabilization with two types
  3. Stabilization model with three types
  4. Main results
  5. Acknowledgments.
  6. Switching position for the excess
  7. Limiting excess distribution
  8. Hitting time bounds

Key Result

Theorem 1.1

For $p\in(0,1)\setminus\{1/2\}$ fixed, we have that in distribution as $n\to\infty$ where the limit is Gaussian with mean $0$ and variance $p(1-p)$. For $p=1/2$, in distribution as $n\to\infty$ where $\chi_3$ is the chi distribution with parameter $3$. For $p=1/2+\lambda/(2\sqrt n)$ with $\lambda\in\mathbb R$ fixed, we have in distribution as $n\to\infty$ where $B_t^\lambda$ is a Brownian motio

Figures (1)

  • Figure 1: An example illustrating the points $M_0=M$, $M_1$, and $K=M_2$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:KU']}
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['prop:EM']}
  • proof : Proof of Theorem \ref{['thm:fixp']}
  • proof : Proof of Theorem \ref{['thm:crit']}
  • ...and 3 more