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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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picakhu
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  • Not sure it makes any difference, but I'll ask anyway: is this line infinite in both directions, or only in the forward direction? – Gerry Myerson Feb 04 '11 at 06:08
  • 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. – picakhu Feb 04 '11 at 06:14
  • NOTE: I have been very vague about the definition of "section", I am open to suggestions about that. – picakhu Feb 04 '11 at 06:24
  • Didn't you already get a counterexample in the comments at SE? – Joel David Hamkins Feb 04 '11 at 10:57
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    +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. – Willie Wong Feb 04 '11 at 11:14
  • Also, could you clarify whether all the movement is synchronized on the same time steps? – Joel David Hamkins Feb 04 '11 at 11:19
  • @JDH, a deterministic model is not stochastic in my opinion, so the counter example is not really a counter example I think. The counter examples are deterministic, and its really difficult to talk about what happens if you move from those cases. As for whether it is synchronized, yes, they all move at once. (coins are tossed prior for all people who can move, and if they are granted permission, all advance at the same time.) – picakhu Feb 04 '11 at 13:17
  • This is the sort of thing which is studied under the term "assymetric exclusion process". I don't know much about it, but that might help get your search started. – David E Speyer Feb 04 '11 at 14:49
  • @David: I do not think this falls under ASEP, but then again, I have not studied them. But you are correct in that the idea of looking for different rates in distinct regions stems from ASEP. – picakhu Feb 05 '11 at 19:55

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