I posted this on [SE][1] and did not get any replies. 

As a recap, there is a sequence of people on a line which has a infinite number of spots. People occupy one spot each. 

If a person is "clear" (which means that the person is at location x, and all three points $x+1$,$x+2$,$x-1$ are vacant) then the person will move forward to $x+1$. Otherwise if the spot $x+1$ is not occupied, the person can move forward to that position with probability $\alpha$. Else, the person is stationary.

Assume that the density of people is more than 0.33 (it can be less too, but I just choose that to avoid the sparsity problem with the initial condition when the movement is deterministic). Now, the question is, can it be shown that there are two distinct regions that form, namely one that is dense and one that is in "free flow" where people keep moving. With the aim to show that eventually all the little dense groups will accumulate together into one large dense section. 


  [1]: https://math.stackexchange.com/questions/20122/infinite-line-of-people