# PushTASEP in inhomogeneous space and its limit shape

### Introduction

Checking that the pushTASEP’s height function in inhomogeneous space is described via the predicted theoretical limit shape from [1].

### The model

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.

### Sampling algorithm

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.

### Data file format

The data files are integer arrays of the form

{{a,b,c,d}}


of length $n$. Each number of the array is an integer $\in \{0,1 \}$, 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

$$h(t,x)=\#\{\text{number of particles which are \le x at time t}\}$$

with $h$ and $x$ rescaled by $t$, and the corresponding limit shape from [1].

## code • (Main GitHub repo)

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

## simulation results

1. ##### Homogeneous case • (data: 4.7 KB) • (image: 20 KB)
$n=2400$, $t=400$, $\xi(x)\equiv 1$
2. ##### Slowdown • (data: 4.7 KB) • (image: 21 KB)
$n=2400$, $t=400$, $\xi(x)=\mathbf{1}_{x<800}+\frac12\cdot\mathbf{1}_{x\ge 800}$
3. ##### Slowdown, height function adjusted by a linear shift to better see fluctuations • (data: 4.7 KB) • (image: 99 KB)
$n=2400$, $t=400$, $\xi(x)=\mathbf{1}_{x<800}+\frac12\cdot\mathbf{1}_{x\ge 800}$
4. ##### Speedup • (data: 4.7 KB) • (image: 18 KB)
$n=2400$, $t=400$, $\xi(x)\mathbf{1}_{x<800}+2\cdot\mathbf{1}_{x\ge 800}$
5. ##### Speedup, height function adjusted by a linear shift to better see fluctuations • (data: 4.7 KB) • (image: 90 KB)
$n=2400$, $t=400$, $\xi(x)\mathbf{1}_{x<800}+2\cdot\mathbf{1}_{x\ge 800}$

## references

1. L. Petrov, In preparation (2018)