Deterministic Random Walks on Regular Trees

TL;DR

This paper investigates the deterministic rotor-router model on regular trees, showing that unlike on grids, deviations from random walk expectations can be arbitrarily large, but require exponentially many vertices to achieve significant deviations.

Contribution

It demonstrates that the rotor-router model on infinite regular trees can produce arbitrarily large deviations from the expected random walk distribution, contrasting previous results on grids.

Findings

Deviations can be arbitrarily large on infinite regular trees.

Achieving large deviations requires exponentially many vertices.

The property of bounded deviation does not hold for all graphs, especially trees.

Abstract

Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid $\Z^d$ and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors. This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite $k$-ary tree ($k \ge 3$), we show that for any deviation $D$ there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least $D$ more chips than expected in the random walk model. However, to achieve a deviation of $D$ it is necessary that at least $\exp(\Omega(D^2))$ vertices contribute by being occupied by a number of chips not divisible by $k$ at a certain time.