TY - JOUR

T1 - Deterministic random walks on the integers

AU - Cooper, Joshua

AU - Doerr, Benjamin

AU - Spencer, Joel

AU - Tardos, Gábor

N1 - Funding Information:
The authors enjoyed the hospitality, generosity and the strong coffee of the Rényi Institute (Budapest) while doing this research. Spencer’s research was partially supported by EU Project Finite Structures 003006; Doerr’s by the “Combinatorial Structure of Intractable Problems” project carried out by the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, in the framework of the European Community’s “Human Resource and Mobility” programme; Cooper’s by an NSF Postdoctoral Fellowship (USA, NSF Grant DMS-0303272); and Tardos’s by the Hungarian National Scientific Research Fund grants OTKA T-046234, AT-048826 and NK-62321.

PY - 2007/11

Y1 - 2007/11

N2 - Jim Propp's P-machine, also known as the 'rotor router model', is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c1, which is approximately 2.29. For intervals of length L, this improves to a difference of O (log L), for the L2 average of a contiguous set of intervals even to O (sqrt(log L)). All these bounds are tight.

AB - Jim Propp's P-machine, also known as the 'rotor router model', is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c1, which is approximately 2.29. For intervals of length L, this improves to a difference of O (log L), for the L2 average of a contiguous set of intervals even to O (sqrt(log L)). All these bounds are tight.

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U2 - 10.1016/j.ejc.2007.04.018

DO - 10.1016/j.ejc.2007.04.018

M3 - Article

AN - SCOPUS:35148825651

VL - 28

SP - 2072

EP - 2090

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 8

ER -