# Deterministic random walks on regular trees

Joshua Cooper, Benjamin Doerr, Tobias Friedrich, Joel Spencer

Research output: Contribution to journalArticle

### 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(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time.

Original language English (US) 353-366 14 Random Structures and Algorithms 37 3 https://doi.org/10.1002/rsa.20314 Published - Oct 2010

### Fingerprint

Random walk
Chip
Vertex of a graph
Deviation
Graph in graph theory
K-tree
Configuration
Model
Routers
Divisible
Router
Walk
Rotor
Rotors
Grid
Analogue
Necessary
Arbitrary

### Keywords

• Discrepancy
• Quasirandomness
• Rotor-router model

### ASJC Scopus subject areas

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

### Cite this

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

In: Random Structures and Algorithms, Vol. 37, No. 3, 10.2010, p. 353-366.

Research output: Contribution to journalArticle

Cooper, Joshua ; Doerr, Benjamin ; Friedrich, Tobias ; Spencer, Joel. / Deterministic random walks on regular trees. In: Random Structures and Algorithms. 2010 ; Vol. 37, No. 3. pp. 353-366.
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