Deterministic Random Walks for Rapidly Mixing Chains

The rotor-router model is a deterministic process analogous to a simple random walk on a graph. This paper is concerned with a generalized model, functional-router model, which imitates a Markov chain possibly containing irrational transition probabilities. We investigate the discrepancy of the number of tokens at a single vertex between the functional-router model and its corresponding Markov chain, and give an upper bound in terms of the mixing time of the Markov chain.