Jim Propp:
Inspired by some joint investigations I conducted with Steve Witham a year or so ago, I recently did a simulation of an even more natural "derandomized random walk" in the quadrant: see http://jamespropp.org/quincunx.gif This shows what happens when you use a non-random Galton board in which the pins are replaced by flip-flops, so that the 1st marble to arrive at a flip-flop gets routed to the right, the 2nd gets routed to the left, the 3rd gets routed to the right, and so on, in alternation.
I've created a "QuasiGalton" rule for exploring this CA in Golly: http://www.trevorrow.com/golly/QuasiGalton.zip If you're using Golly 2.1 you can open the .zip in Golly and it will automatically install QuasiGalton.tree and QuasiGalton.colors in your rules folder, then run build-galton.py to construct a Galton board pattern (this takes a few secs). The .zip also includes QuasiGalton.txt which contains the Python transition function used to create QuasiGalton.tree via the make-ruletree.py script which can be downloaded from the Rule Table Repository: http://ruletablerepository.googlecode.com/files/make-ruletree.py (This script will be included in Golly 2.2. It makes it very easy to create new rules.) Andrew