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:
 Distance parameter $L>0$ (in asymptotic regimes we consider this parameter will grow to infinity);
 Speed function $\xi(y),$ $y\in \mathbb R_{\geq 0},$ which is assumed to be positive, piecewise continuous,
have left and right limits, and uniformly bounded away from $0$ and $+\infty;$
 Discrete set $\mathbf{B}\subset \mathbb R_{>0}$
without
accumulation points
and such that there are only finitely many points of $\mathbf{B}$
in a right neighborhood of $0$.
Fix a function $p: \mathbf B\rightarrow (0,1)$.
The process $X(t)$ evolves as follows:
 New particles
enter $\mathbb R_{>0}$ (leaving $0$) at
rate
$\xi(0)$
(we say
that a certain event has rate $\mu>0$ if it repeats after independent random
time intervals
which have exponential distribution with rate $\mu$ (and mean
$\mu^{1}$));
 If at some time $t>0$ there are particles at a location $x \in R_{>0}$,
then one particle decides to leave this location at rate $\xi(x)$
(these events occur independently for each occupied location). Almost surely at each
moment in time only one particle can start moving;
 The moving particle (say, $x_j$) instantaneously jumps to the right by some random
distance $x_j(t)x_j(t)=\min(Y, x_{j1}(t)x_j(t))$ (by agreement, $x_0\equiv+\infty$).
The distribution of $Y$ is as follows:
$
Prob
(Y \geq y )
=
e^{L y}\prod\limits_{b \in \mathcal{\mathbf B},
\text{ }x_j(t)<b<x_j(t)+y} p(b).
$
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.
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.
(note: parameters in the code might differ from the ones in simulation results below)
simulation results

$t=1000, \xi(x)\equiv 1$

Fully homogeneous case with limit shape • (data: 15 KB) • (image: 16 KB)
$t=1000, \xi(x)\equiv 1$

$t=1000, \xi(x)=\mathbf{1}_{x<0.1}+\frac12\mathbf{1}_{x\ge 0.1}$

Mild slowdown (about critical), zoom around the slowdown • (data: 16 KB) • (image: 13 KB)
$t=1000, \xi(x)=\mathbf{1}_{x<0.1}+\frac12\mathbf{1}_{x\ge 0.1}$

$t=1000, \xi(x)=\mathbf{1}_{x<0.1}+0.3\cdot\mathbf{1}_{x\ge 0.1}$

Harder slowdown, zoom around the slowdown • (data: 15 KB) • (image: 12 KB)
$t=1000, \xi(x)=\mathbf{1}_{x<0.1}+0.3\cdot\mathbf{1}_{x\ge 0.1}$

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), 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}
$
references

A. Knizel, L. Petrov, A. Saenz. In preparation (2018)

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