Rotor Walks and Markov Chains

TL;DR

This paper studies rotor walks, a deterministic analogue of random walks, showing they closely mimic the probabilistic behavior with stronger concentration results on hitting times and frequencies.

Contribution

It extends rotor walk analysis to Markov chains, proving concentration of key quantities and demonstrating they outperform repeated random walk simulations in accuracy.

Findings

Normalized hitting frequencies concentrate around expected values.

Discrepancy after n runs is at most C/n, better than c/sqrt n.

Rotor walks provide a deterministic alternative with strong probabilistic guarantees.

Abstract

The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to Markov chains on a countable state space. Subject to general conditions, we prove that many natural quantities associated with the rotor walk (including normalized hitting frequencies, hitting times and occupation frequencies) concentrate around their expected values for the random walk. Furthermore, the concentration is stronger than that associated with repeated runs of the random walk, with discrepancy at most C/n after n runs (for an explicit constant C), rather than c/sqrt n.