Does anyone know of any work on the following model or variants thereof?:
Finitely many chips are distributed on the integers at time 0. To find the distribution at time $t+1$, take all the chips at each location $n$ at time $t$ and send equal numbers of them to $n-1$ and $n+1$, with the proviso that if the numbers of chips at $n$ at time $t$ was odd, say $2k+1$, then $k$ chips go left, $k$ chips go right, and the last chip goes either left or right, as determined by a coin flip. (We assume that the coin used at location $n$ at time $t$ is independent of all the coin flips at all other locations and times.)
Note that if the ``odd chip'' is required to stay put at $n$, this is precisely the chip-firing or sandpile analogue of random walk on the integers, as described for instance in R. J. Anderson, L. Lovasz, P. W. Shor, J. Spencer, E. Tardos and S. Winograd, Disks, balls and walls: Analysis of a combinatorial game, American Math. Monthly 96, pp. 481-493 (1989).