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Abstract:
Jim Propp’s P-machine, also known as the ‘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(logL), for the L2 average of a contiguous
set of intervals even to . All these bounds are tight.
