Continuous space TASEP

Leonid Petrov


The model [1]

The continuous space TASEP is a continuous time Markov process on the space

$$ \mathcal{X}:=\{(x_1\geq x_2\ge\dots \geq x_k>0) \colon x_i\in \mathbb R \text{ and } k\in \mathbb Z_{\geq 0} \text{ is arbitrary}\} $$

of finite particle configurations on $\mathbb R_{>0}.$ The particles are ordered, and the process preserves this ordering. However, more than one particle per site it allowed.

The Markov process $X(t)$ on $\mathcal{X}$ depends on the following data:

The process $X(t)$ evolves as follows:

This model is a $q=0$ degeneration of the one studied in [2].

Sampling algorithm

The simulation is straightforward. In continuous time, we choose which particle jumps, then increase the continuous time count by the corresponding exponential random variable, and perform the necessary comparisons.

Data file format

The data files are integer arrays like

{0.0286274477178, 0.0286274477178, 0.210833155189, }

of particle locations. The plots display the height function (rescaled) which counts the number of particles to the right of a given location. In some plots the theoretical limit shape is also given.


code • (Main GitHub repo)

(note: parameters in the code might differ from the ones in simulation results below)

simulation results

  1. Fully homogeneous case • (data: 15 KB) • (image: 14 KB)
    $t=1000, \xi(x)\equiv 1$
    Fully homogeneous case
  2. Fully homogeneous case with limit shape • (data: 15 KB) • (image: 16 KB)
    $t=1000, \xi(x)\equiv 1$
    Fully homogeneous case with limit shape
  3. Mild slowdown (about critical) • (data: 16 KB) • (image: 17 KB)
    $t=1000, \xi(x)=\mathbf{1}_{x<0.1}+\frac12\mathbf{1}_{x\ge 0.1}$
    Mild slowdown (about critical)
  4. Mild slowdown (about critical), zoom around the slowdown • (data: 16 KB) • (image: 13 KB)
    $t=1000, \xi(x)=\mathbf{1}_{x<0.1}+\frac12\mathbf{1}_{x\ge 0.1}$
    Mild slowdown (about critical), zoom around the slowdown
  5. Harder slowdown • (data: 15 KB) • (image: 16 KB)
    $t=1000, \xi(x)=\mathbf{1}_{x<0.1}+0.3\cdot\mathbf{1}_{x\ge 0.1}$
    Harder slowdown
  6. Harder slowdown, zoom around the slowdown • (data: 15 KB) • (image: 12 KB)
    $t=1000, \xi(x)=\mathbf{1}_{x<0.1}+0.3\cdot\mathbf{1}_{x\ge 0.1}$
    Harder slowdown, zoom around the slowdown
  7. Slower initial stream of particles (roadblock in the beginning) • (data: 9 KB) • (image: 15 KB)
    Gaussian behavior for $0 < x < 0.04$. Dashed curve is the limit shape with full initial stream of particles.
    $t=1000, \xi(x)= 0.6\cdot\mathbf{1}_{x=0}+\mathbf{1}_{0 < x<0.1}+0.3\cdot\mathbf{1}_{x\ge 0.1} $
    Slower initial stream of particles (roadblock in the beginning)
  8. Slower initial stream of particles (roadblock in the beginning), zoom around the Gaussian behavior • (data: 9 KB) • (image: 13 KB)
    Gaussian behavior for $0 < x < 0.04$. Dashed curve is the limit shape with full initial stream of particles.
    $t=1000, \xi(x)= 0.6\cdot\mathbf{1}_{x=0}+\mathbf{1}_{0 < x<0.1}+0.3\cdot\mathbf{1}_{x\ge 0.1} $
    Slower initial stream of particles (roadblock in the beginning), zoom around the Gaussian behavior

references

  1. A. Knizel, L. Petrov, A. Saenz. In preparation (2018)
  2. A. Borodin, L. Petrov, Inhomogeneous exponential jump model (2017), Probability Theory and Related Fields, to appear • https://arxiv.org/abs/1703.03857