Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact solution of TASEP and variants with inhomogeneous speeds and memory lengths

Konstantin Matetski, Daniel Remenik

Abstract

In [arXiv:1701.00018, arXiv:2107.07984] an explicit biorthogonalization method was developed that applies to a class of determinantal measures which describe the evolution of several variants of classical interacting particle systems in the KPZ universality class. The method leads to explicit Fredholm determinant formulas for the multipoint distributions of these systems which are suitable for asymptotic analysis. In this paper we extend the method to a broader class of determinantal measures which is applicable to systems where particles have different jump speeds and different memory lengths. As an application of our results we study three particular examples: some variants of TASEP with two blocks of particles having different speeds, a version of discrete time TASEP which mixes particles with sequential and parallel update, and a version of sequential TASEP with a block of long memory particles placed at the bulk of the system.

Exact solution of TASEP and variants with inhomogeneous speeds and memory lengths

Abstract

In [arXiv:1701.00018, arXiv:2107.07984] an explicit biorthogonalization method was developed that applies to a class of determinantal measures which describe the evolution of several variants of classical interacting particle systems in the KPZ universality class. The method leads to explicit Fredholm determinant formulas for the multipoint distributions of these systems which are suitable for asymptotic analysis. In this paper we extend the method to a broader class of determinantal measures which is applicable to systems where particles have different jump speeds and different memory lengths. As an application of our results we study three particular examples: some variants of TASEP with two blocks of particles having different speeds, a version of discrete time TASEP which mixes particles with sequential and parallel update, and a version of sequential TASEP with a block of long memory particles placed at the bulk of the system.
Paper Structure (33 sections, 18 theorems, 252 equations, 1 figure)

This paper contains 33 sections, 18 theorems, 252 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivating examples: interacting particle systems
  3. Continuous time TASEP
  4. Discrete time TASEPs with right Bernoulli jumps
  5. Caterpillars with right Bernoulli jumps
  6. Other types of caterpillars
  7. PushASEP
  8. Biorthogonal ensemble formula for determinantal measures
  9. Explicit biorthogonalization scheme
  10. Setting
  11. The biorthogonalization problem
  12. The boundary value problem
  13. Main result: representation in terms of random walk hitting times
  14. Preliminaries
  15. Main result for the kernel
  16. ...and 18 more sections

Key Result

Proposition 2.1

The distribution function of $X_t=X^{{\text{\upshape{head}}}}_t$ for continuous time TASEP is given by TASEP-different_speeds-revision-v2-shortened-revised:eq:caterpillars/TASEP-different_speeds-revision-v2-shortened-revised:eq:caterpillars-Kt with $\varphi(w) = e^w$ and $L_i = 1$ for all $i \in \ll

Figures (1)

  • Figure 1: The contour $\mathcal{C}=\mathcal{C}_1 \cup \mathcal{C}_2$ chosen for the steep descent analysis. $\mathcal{C}_1$ is the red part of the two rays departing off the origin, while $\mathcal{C}_2$ is the blue arc of the circle $\tilde{\Gamma}$. The radius $R$ of the circle and the coordinate $R_0$ of its center can be computed based on the length $\ell$ of the left side of the triangle and the values of the two bottom angles $\phi$ and $\psi$.

Theorems & Definitions (30)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • Theorem 3.3
  • Remark 4.3
  • Lemma 4.4
  • ...and 20 more