Electronic Communications in Probability

Transience and recurrence of rotor-router walks on directed covers of graphs

Wilfried Huss and Ecaterina Sava

Full-text: Open access

Abstract

The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question on directed covers of finite graphs, which are also called trees with finitely many cone types or periodic trees. Furthermore, we provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree. <strong><a href="http://dx.doi.org/10.1214/ECP.v19-3848">An Erratum is available in ECP volume 19, paper 71, (2014)</a>.</strong>

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 41, 13 pp.

Dates
Accepted: 22 September 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263174

Digital Object Identifier
doi:10.1214/ECP.v17-2096

Mathematical Reviews number (MathSciNet)
MR2981897

Zentralblatt MATH identifier
1252.05035

Subjects
Primary: 05C05: Trees
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Keywords
graphs directed covers rotor-router walks multitype branching process recurrence transience

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Huss, Wilfried; Sava, Ecaterina. Transience and recurrence of rotor-router walks on directed covers of graphs. Electron. Commun. Probab. 17 (2012), paper no. 41, 13 pp. doi:10.1214/ECP.v17-2096. https://projecteuclid.org/euclid.ecp/1465263174


Export citation

References

  • Angel, Omer; Holroyd, Alexander E. Rotor walks on general trees. SIAM J. Discrete Math. 25 (2011), no. 1, 423–446.
  • Bak, Per; Tang, Chao; Wiesenfeld, Kurt. Self-organized criticality. Phys. Rev. A (3) 38 (1988), no. 1, 364–374.
  • Cooper, Joshua; Doerr, Benjamin; Spencer, Joel; Tardos, Garbor. Deterministic random walks. Proceedings of the Eighth Workshop on Algorithm Engineering and Experiments and the Third Workshop on Analytic Algorithmics and Combinatorics, 185–197, SIAM, Philadelphia, PA, 2006.
  • Cooper, Joshua N.; Spencer, Joel. Simulating a random walk with constant error. Combin. Probab. Comput. 15 (2006), no. 6, 815–822.
  • Doerr, Benjamin; Friedrich, Tobias. Deterministic random walks on the two-dimensional grid. Algorithms and computation, 474–483, Lecture Notes in Comput. Sci., 4288, Springer, Berlin, 2006.
  • Giacaglia, Giuliano Pezzolo; Levine, Lionel; Propp, James; Zayas-Palmer, Linda. Local-to-global principles for the hitting sequence of a rotor walk. Electron. J. Combin. 19 (2012), no. 1, P5, 23 pp.
  • Gilch, Lorenz A.; MĂźller, Sebastian. Random walks on directed covers of graphs. J. Theoret. Probab. 24 (2011), no. 1, 118–149.
  • Harris, Theodore E. The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119 Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J. 1963 xiv+230 pp.
  • Holroyd, Alexander E.; Propp, James. Rotor walks and Markov chains. Algorithmic probability and combinatorics, 105–126, Contemp. Math., 520, Amer. Math. Soc., Providence, RI, 2010.
  • Huss, Wilfried; Sava, Ecaterina. Rotor-router aggregation on the comb. Electron. J. Combin. 18 (2011), no. 1, P224, 23 pp.
  • Huss, Wilfried; Sava, Ecaterina. The rotor-router group of directed covers of graphs, Electron. J. Combin. 19 (2012), no. 3, P30.
  • Landau, Itamar; Levine, Lionel. The rotor-router model on regular trees. J. Combin. Theory Ser. A 116 (2009), no. 2, 421–433.
  • R. Lyons and Y. Peres, Probabilty on trees and networks, preprint.
  • Lyons, Russell. Random walks and percolation on trees. Ann. Probab. 18 (1990), no. 3, 931–958.
  • Nagnibeda, Tatiana; Woess, Wolfgang. Random walks on trees with finitely many cone types. J. Theoret. Probab. 15 (2002), no. 2, 383–422.
  • V. B. Priezzhev, Deepak Dhar, Abhishek Dhar, and Supriya Krishnamurthy, Eulerian walkers as a model of self-organized criticality, Phys. Rev. Lett. 77 (1996), no. 25, 5079–5082.
  • Takacs, Christiane. Random walk on periodic trees. Electron. J. Probab. 2 (1997), no. 1, 1–16 (electronic).