This is a simulation of the simple exclusion in i.i.d. random environment. Intuitively, this process is described as follows: We have a segment of size N and k particles. We place the particles on the segment such that each site is occupied by at most one particle. Moreover, we sample an i.i.d. random environment, i.e. for every site x assign an i.i.d. random variable ω(x). We keep the environment fixed over time and let the particles move as follows: At each time step, choose a vertex x uniformly at random. If x is occupied by a particle, then this particle tries to move to the right with probability ω(x) and to the left with probability 1-ω(x). However, if the target site is already occupied, this move is suppressed.

The transition rates of the simple exclusion process in i.i.d. random environment are visualized as follows:

In the simulation, you are free to choose the size of the segment N and the number of particles k. For the environment law, you can choose either the Uniform-distribution on a subinterval of [ε,1] (where ε=0.01) or a two coin law, i.e. a mixture of two Dirac measure on [ε,1]. The environment is visualized in the top row, where purple indicated drift to the left and green a dirft to the right. Moreover, there is the option of running two simple exclusion processes within the canonical coupling. This means that the same vertex is chosen to be updated in both dynamics and that the same environment is used when both processes have the same environment law. The corresponding disagreement process is shown in the third row. All first class particles are shown in red, while a disagreement is drawn in blue. Depending on the choice of parameters, there are various different regimes and we give examples for some of the regimes. Since the processes take in some cases a rather long time to couple, there is the option to do N simulations at once. Also, you can show and hide the height function representation and the boxes in the segment.

Enough for the description - Have fun with the simulation :)