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Thesis

#### Deterministic Random Walks on Infinite Grids

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##### Citation

Friedrich, T. (2005). Deterministic Random Walks on Infinite Grids. Master Thesis, Friedrich-Schiller-Universität Jena, Saarbrücken.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-2639-D

##### Abstract

The rotor-router model is a simple deterministic analogue of random walk invented by Jim Propp. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order. This thesis investigates how well this process simulates a random walk on an infinite two-dimensional grid. 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 $c$. It is proved that $7.2 < c < 11.8$ in general.
Surprisingly, these bounds depend on the order in which the neighbors are served. It is also shown that in a generalized setting, where one just requires that no neighbor gets more than $\Delta$ chips more than another, there is also such a constant $c'$ with $7.7\Delta < c' < 26.9\Delta$.