PushTASEP in inhomogeneous space and its limit shape

Leonid Petrov


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 , 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

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

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

references

  1. L. Petrov, In preparation (2018)