Let us assume that we have an infinite line of people, and each person can either move forwards or remain at the same place. They move only one step at a time. (They are jumping from one position to the next if that position is empty). All people either move or remain still simultaneously.
Let us further assume that a person in spot $x$ will move forward to spot $x+1$ if spots $x+1$,$x+2$ and $x-1$ are empty. For all other people, they move with a probability $\alpha$.
Can it be shown that "eventually", there will be only two regions, one where the density is high and one where the density is low? (the fast lane, and a jammed lane)
A way to think about this is to think about the people being on a finite circular loop. (obviously then there are not infinite people in the line)
EDIT:
For the purposes of clarification since many people are worried about the initial configuration, if it is assumed that it is sufficiently dense. (more than 0.33). We can have any initial configuration, it should not matter as we are interested in long term behavior.
EDIT 2:
The objection raised below in the comments is deterministic. However, the question is about stochastic, thus we can say that $\alpha \not = 0$ or $1$.
Example: Let 1 be a occupied spot and a 0 an unoccupied spot. then we can start with
...1110001000...
and then next we have either
...1110000100...
or
...1101000100...
depending on whether the person advanced who had the probability $\alpha$ of advance.
EDIT: C++ PROGRAM
Program that I wrote. File You may have to add a .exe to the end if you want to run it in Windows OS.
Same program with colors fixed and much longer time frame. File2