Deterministic random walks on regular trees

Authors:
Joshua Cooper;Benjamin Doerr;Tobias Friedrich;Joel Spencer
Affiliations:
University of South Carolina, Columbia, SC;Algorithms and Complexity, Max-Planck-Institut für Informatik, Saarbrücken, Germany;Algorithms and Complexity, Max-Planck-Institut für Informatik, Saarbrücken, Germany;Courant Institute of Mathematical Sciences, New York University, New York, NY
Venue:
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2008

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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 Zd and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips 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 ≥ 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 kΘ(D) vertices contribute by being occupied by a number of chips not divisible by k in a certain time interval.