PushTASEP in inhomogeneous space and its limit shape     TASEPs

Simulation of a single-layer PushTASEP in inhomogeneous space, comparing particle evolution to theoretical limit shape predictions. The visualization shows the height function of the particle system with rescaling to reveal macroscopic behavior.

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

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

code

• Link to code (python2 for simulations, simple Mathematica for drawing. Mathematica source not present)

1. Homogeneous case • (data: 4.7 KB) • (graphics: 20 KB)

   $n=2400$, $t=400$, $\xi(x)\equiv 1$
   

2. Slowdown • (data: 4.7 KB) • (graphics: 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) • (graphics: 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) • (graphics: 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) • (graphics: 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)

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