Rotor walks on general trees
Omer Angel, Alexander E. Holroyd

TL;DR
This paper studies rotor walks on infinite trees, showing their escape behavior aligns with random walk probabilities under certain conditions, and reveals phase transitions and bounds on escape discrepancies.
Contribution
It establishes the relationship between rotor walk escape proportions and random walk probabilities on trees, including phase transition phenomena and bounds on discrepancies.
Findings
Proportion of escapes equals random walk escape probability under certain conditions.
Discontinuous phase transition in escape probability for i.i.d. initial rotor directions.
Existence of trees with arbitrarily large escape discrepancy.
Abstract
The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially towards the root. For i.i.d. random initial rotor directions on a regular tree, the limiting proportion of escapes is either zero or the random walk escape probability, and undergoes a discontinuous phase transition between the two as the distribution is varied. In the critical case there are no escapes, but the walker's maximum distance from the root grows doubly exponentially with the number of visits to the…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsStochastic processes and statistical mechanics · Theoretical and Computational Physics · Markov Chains and Monte Carlo Methods
