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The second class particle process at shocks

Patrik L. Ferrari, Peter Nejjar

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities $λ$ to the left of the origin and $ρ$ to the right of it and $λ<ρ$. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.

The second class particle process at shocks

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities to the left of the origin and to the right of it and . We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.
Paper Structure (13 sections, 24 theorems, 85 equations, 5 figures)

This paper contains 13 sections, 24 theorems, 85 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Distribution of the second class particle
  3. The model
  4. Distribution of the second class particle
  5. Asymptotic results
  6. Large time results
  7. Backwards paths
  8. Slow decorrelation
  9. Proof of Theorem \ref{['thmMain']}
  10. Proof of Corollary \ref{['truecor']}
  11. Limit process for flat IC
  12. A bound for step initial conditions
  13. Acknowledgements.

Key Result

Theorem 2.1

With the above notations, for $0\leq t_1<t_2<\ldots<t_m$ and $x_1,\ldots,x_n\in\mathbb{Z}$, we have the identity

Figures (5)

  • Figure 1: The initial height function $h(x,0)$ is the dashed thick line, which matches with $h^-(x,0)$ for $x<0$ and with $h^+(x,0)$ for $x>0$. The second class particle starts at $x=0$. The small vertical shifts are just for making the illustration clearer and avoid overlapping height functions with different initial conditions.
  • Figure 2: Illustration of the configurations and height function. A clock rings at position $Y(\tau^-)$ at time $\tau$. (Left): the second class particle jumps to the right; (Right): the second class particle jumps to the left.
  • Figure 3: Construction of the backwards path as in Definition \ref{['DefinBackwardsPaths']}. A Poisson event happens at $(x(s),s)$. The height function at time $s$ is in blue, while the height function just before is in black.
  • Figure 4: Localization strategy: (left) with the same initial conditions, geodesics to $B_-$ are to the left of the right-most geodesics to the point $D=(v_s (t-t^\nu),t-t^\nu)$, which by hypothesis is to the left of $-\tfrac{1}{2}\delta t$ with high probability. (right) if the geodesics to $B_-$ is to the left of $-\tfrac{1}{2}\delta t$, then it is to the left of the right-most one with step initial condition to $B_-$, which by hypothesis is not fluctuating more than $\mathcal{O}(t^{2/3})$, thus it stays to the left of the line $(0,0)$ to $D$.
  • Figure 5: The setting for the application of slow decorrelation. The red line is the macroscopic trajectory of the shock.

Theorems & Definitions (43)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 33 more