Checking that the pushTASEP’s height function in inhomogeneous space is described via the predicted theoretical limit shape from [1].
Fix a positive function $\xi(x)$, $x\in\mathbb{Z}_{\ge0}$, separated from $0$ and $\infty$. We consider the pushTASEP in inhomogeneous space with $\xi(x)$ playing the role of speed. Namely, starting from the step initial configuration ${1,2,\ldots}$, each particle at every location $x$ has independent exponential clock with rate $\xi(x)$ and mean waiting time $1/\xi(x)$. If the clock rings, the particle jumps to the right by one, also pushing to the right by one the whole packed cluster of particles immediately to the right of it.
The simulation is straightforward. In fact, I reused the multilayer simulation from a previous post and simply put the number of layers to be one.
The data files are integer arrays of the form
{{a,b,c,d}}
of length $n$. Each number of the array is an integer , where $1$ means that there is a particle at the corresponding site, and $0$ means the absence of such particle.
The plots display the rescaled height function
with $h$ and $x$ rescaled by $t$, and the corresponding limit shape from [1].
https://github.com/lenis2000/simulations/blob/master/2018-02-10-PushTASEP-single/2018-02-10-PushTASEP-single.py
(python2 for simulations, simple Mathematica for drawing. Mathematica source not present)