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I posted this on SE 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.

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  • $\begingroup$ Not sure it makes any difference, but I'll ask anyway: is this line infinite in both directions, or only in the forward direction? $\endgroup$
    – Gerry Myerson
    Commented Feb 4, 2011 at 6:08
  • $\begingroup$ Erm, think about it as a finite loop that has spots added to it. We are in the limiting case when there are infinitely many spots. $\endgroup$
    – picakhu
    Commented Feb 4, 2011 at 6:14
  • $\begingroup$ NOTE: I have been very vague about the definition of "section", I am open to suggestions about that. $\endgroup$
    – picakhu
    Commented Feb 4, 2011 at 6:24
  • $\begingroup$ Didn't you already get a counterexample in the comments at SE? $\endgroup$
    – Joel David Hamkins
    Commented Feb 4, 2011 at 10:57
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    $\begingroup$ +1 JDH. @picakhu: what exactly is the point of asking for my opinion on MSE when you just forge ahead anyway? Your current phrasing of the question does not rule out the various edge cases that JDH and I raised. $\endgroup$
    – Willie Wong
    Commented Feb 4, 2011 at 11:14

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