[HTML][HTML] Deterministic random walks on the integers
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 …
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 …
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 O(logL). All these bounds are tight.
Elsevier