Abstract
We consider the model of the one-dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite. We give a criterion to decide whether the limiting speed of the walk is non-zero. In particular, we show that a positive speed may be obtained for just three cookies per site. We also prove a result on the continuity of the speed with respect to the initial cookie distribution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Antal T. and Redner S. (2005). The excited random walk in one dimension. J. Phys. A 38(12): 2555–2577
Basdevant, A.-L., Singh, A.: Rate of growth of a transient cookie random walk, 2007. Preprint. Available via http://arxiv.org/abs/math.PR/0703275
Benjamini I. and Wilson D.B. (2003). Excited random walk. Electron. Commun. Probab. 8: 86–92
Davis B. (1999). Brownian motion and random walk perturbed at extrema. Probab. Theory Relat. Fields 113(4): 501–518
Feller W. (1971). An introduction to probability theory and its applications, vol. II. Wiley, New York
Kesten H., Kozlov M.V. and Spitzer F. (1975). A limit law for random walk in a random environment. Compositio Math. 30: 145–168
Kozma, G.: Excited random walk in three dimensions has positive speed, 2003. Preprint. Available via http://arxiv.org/abs/math.PR/0310305
Kozma, G.: Excited random walk in two dimensions has linear speed, 2005. Preprint. Available via http://arxiv.org/abs/math.PR/0512535
Mountford T., Pimentel L.P.R. and Valle G. (2006). On the speed of the one-dimensional excited random walk in the transient regime. Alea 2: 279–296, (electronic)
Norris J.R. (1998). Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2. Cambridge University Press, Cambridge, Reprint of 1997 original
Vatutin V.A. and Zubkov A.M. (1993). Branching processes. II. J. Soviet Math. 67(6): 3407–3485, Probability theory and mathematical statistics, 1
Vinokurov. G.V.: On a critical Galton–Watson branching process with emigration. Teor. Veroyatnost. i Primenen. (English translation: Theory Probab. Appl. 32(2), 351–352 (1987)), 32(2), 378–382 (1987)
Zerner M.P.W. (2005). Multi-excited random walks on integers. Probab. Theory Relat. Fields 133(1): 98–122
Zerner M.P.W. (2006). Recurrence and transience of excited random walks on \(\mathbb{Z}^d\) and strips Electron. Commun. Probab. 11: 118–128, (electronic)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Basdevant, AL., Singh, A. On the speed of a cookie random walk. Probab. Theory Relat. Fields 141, 625–645 (2008). https://doi.org/10.1007/s00440-007-0096-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0096-8