Deterministic random walks on the integers
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Deterministic random walks on regular trees
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The rotor--router model on regular trees
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Deterministic random walks on the two-dimensional grid
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Deterministic random walks on the two-dimensional grid
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Operations Research Letters
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Deterministic walks with choice
Discrete Applied Mathematics
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We analyse Jim Propp's $P$-machine, a simple deterministic process that simulates a random walk on ${\mathbb Z}^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,{\bf x}) : {\bf x} \in {\mathbb Z}^d\}$, where $p(n,{\bf x})$ is the probability that a walk beginning from the origin arrives at ${\bf x}$ at time $n$.