### Abstract

Jim Propp's rotor-router model is a deterministic analog 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 15 (2006) 815-822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid Zdbl;^{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 ≥ 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(Ω(D^{2})) vertices contribute by being occupied by a number of chips not divisible by k at a certain time.

Original language | English (US) |
---|---|

Pages (from-to) | 353-366 |

Number of pages | 14 |

Journal | Random Structures and Algorithms |

Volume | 37 |

Issue number | 3 |

DOIs | |

State | Published - Oct 2010 |

### Fingerprint

### Keywords

- Discrepancy
- Quasirandomness
- Rotor-router model

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*37*(3), 353-366. https://doi.org/10.1002/rsa.20314

**Deterministic random walks on regular trees.** / Cooper, Joshua; Doerr, Benjamin; Friedrich, Tobias; Spencer, Joel.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 37, no. 3, pp. 353-366. https://doi.org/10.1002/rsa.20314

}

TY - JOUR

T1 - Deterministic random walks on regular trees

AU - Cooper, Joshua

AU - Doerr, Benjamin

AU - Friedrich, Tobias

AU - Spencer, Joel

PY - 2010/10

Y1 - 2010/10

N2 - Jim Propp's rotor-router model is a deterministic analog 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 15 (2006) 815-822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid Zdbl;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 ≥ 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(Ω(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time.

AB - Jim Propp's rotor-router model is a deterministic analog 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 15 (2006) 815-822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid Zdbl;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 ≥ 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(Ω(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time.

KW - Discrepancy

KW - Quasirandomness

KW - Rotor-router model

UR - http://www.scopus.com/inward/record.url?scp=77957307545&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957307545&partnerID=8YFLogxK

U2 - 10.1002/rsa.20314

DO - 10.1002/rsa.20314

M3 - Article

AN - SCOPUS:77957307545

VL - 37

SP - 353

EP - 366

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -