Hostname: page-component-77f85d65b8-6bnxx Total loading time: 0 Render date: 2026-03-28T05:12:35.829Z Has data issue: false hasContentIssue false
  • English
  • Français

Deterministic Random Walks on the Two-Dimensional Grid

Published online by Cambridge University Press:  01 March 2009

BENJAMIN DOERR  and
TOBIAS FRIEDRICH
BENJAMIN DOERR
Affiliation:
Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany (e-mail: Tobias.Friedrich@mpi-inf.mpg.de)
TOBIAS FRIEDRICH
Affiliation:
Max-Planck-Institut für Informatik, 66123 Saarbrücken, Germany (e-mail: Tobias.Friedrich@mpi-inf.mpg.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Cooper, J. and Spencer, J. (2006) Simulating a random walk with constant error. Combin. Probab. Comput. 15 815–822.CrossRefGoogle Scholar
[2]Cooper, J., Doerr, B., Spencer, J. and Tardos, G. (2007) Deterministic random walks on the integers. Europ. J. Combin. 28 2072–2090.CrossRefGoogle Scholar
[3]Cooper, J., Doerr, B., Friedrich, T. and Spencer, J. (2008) Deterministic random walks on regular trees. In Proc. 19th ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 766–772.Google Scholar
[4]Diaconis, P. and Fulton, W. (1990) A growth model, a game, an algebra, {L}agrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Pol. Torino 49 95–119.Google Scholar
[5]Kleber, M. (2005) Goldbug variations. The Mathematical Intelligencer 27 55–63.CrossRefGoogle Scholar
[6]Lawler, G. F. (1995) Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23 71–86.CrossRefGoogle Scholar
[7]Lawler, G. F., Bramson, M. and Griffeath, D. (1992) Internal diffusion limited aggregation. Ann. Probab. 20 2117–2140.CrossRefGoogle Scholar
[8]Levine, L. and Peres, Y. (2008) Spherical asymptotics for the rotor–router model in . Indiana Univ. Math. J., no. 1, 431–450.Google Scholar
[9]Levine, L. and Peres, Y. (2005) The rotor–router shape is spherical. The Mathematical Intelligencer 27 9–11.CrossRefGoogle Scholar
[10]Levine, L. and Peres, Y. (2007) Strong spherical asymptotics for rotor–router aggregation and the divisible sandpile. Submitted Potential Analysis; preliminary version available from arXiv:math/07040688.Google Scholar
[11]Moore, C. and Machta, J. (2000) Internal diffusion-limited aggregation: Parallel algorithms and complexity. J. Statist. Phys. 99 661–690.CrossRefGoogle Scholar
[12]Yap, C. K. (2000) Fundamental Problems of Algorithmic Algebra, Oxford University Press.Google Scholar