This model is a version of the one considered in the previous simulation. However, the asymptotic behavior is quite different. The model in this post is related to the column RSK and Schur measures and is amenable to asymptotic analysis via exact formulas.
Fix a positive function $\xi(x)$, $x\in\mathbb{Z}_{\ge0}$, separated from $0$ and $\infty$. Fix a number of layers $k$, and consider the following $k$-layer particle configuration $x^{(j)}_1<x^{(j)}_2<x^{(j)}_3<\ldots$, $j=1,\ldots,k$. This particle configuration evolves in continuous time. The initial condition is the densely packed (step) one,
\[x_i^{(j)}=i,\qquad i=1,2,\ldots,\qquad j=1,\ldots,k.\]The evolution is as follows. At each site $y\in\mathbb{Z}_{\ge0}$ there is an independent exponential clock with rate $\xi(y)$ (so, mean waiting time $1/\xi(y)$). This rate does not depend on the layer’s number. When the clock at site $y$ rings:
The dynamics on the first layer is simply the PushTASEP (= long-range TASEP) studied, e.g., in [1]. The space-inhomogeneous version described above is studied in [2].
To sample the system, we choose a fixed size $n$ (so we consider the behavior on ), the number of layers $k$, and the inhomogeneity function $\xi(\cdot)$.
Since the dynamics is a continuous time Markov process, we sample it directly, using exponential clocks. In more detail, we sample an independent exponential random variable with mean $1/\xi(y)$ for each $y=1,\ldots,n$. Then we choose the minimal of these waiting times, and
The data files are Mathematica readable 2d integer arrays of the form
{{a,b,c,d},{e,f,g,h},{x,y,z,t}}
where $k$ is the number of blocks ($3$ in the above example), and $n$ is the length of a block ($4$ in the above example). Each element of the array is an integer , where $1$ means that there is a particle at the corresponding site and the corresponding layer, and $0$ means the absence of such particle. In other words, the data reads the configuration layer by layer, starting from the first layer, and shows occupation variables.
There are no spaces or line breaks in the file.
The plots show the first layer on top, layers correspond to horizontals. Black squares mean particles, and white squares mean empty space.
https://github.com/lenis2000/simulations/blob/master/2017-12-30-PushTASEP-colRSK/2017-12-30-PushTASEP-colRSK.py
(python2, both for simulation and drawing)
https://arxiv.org/abs/0707.2813
• http://emis.ams.org/journals/EJP-ECP/article/download/541/541-1801-1-PB.pdf
https://arxiv.org/abs/1309.4825
• https://link.springer.com/article/10.1007%2Fs11005-014-0696-z