Deterministic random walks

Jim Propp's P-machine, also known as 'rotor router model' is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant C₁, which is approximately 2.29. For intervals of length L, this improves to a difference of O(log L) (instead of 2.29L), for the L₂ average of a contiguous set of intervals even to O(√log L). It seems plausible that similar results hold for higher-dimensional grids ℤ^d instead of the path ℤ.