# zbMATH — the first resource for mathematics

On the speed of a cookie random walk. (English) Zbl 1141.60383
Summary: 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F15 Strong limit theorems
Full Text:
##### References:
 [1] Antal T. and Redner S. (2005). The excited random walk in one dimension. J. Phys. A 38(12): 2555–2577 · Zbl 1113.82024 [2] 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 · Zbl 1191.60107 [3] Benjamini I. and Wilson D.B. (2003). Excited random walk. Electron. Commun. Probab. 8: 86–92 · Zbl 1060.60043 [4] Davis B. (1999). Brownian motion and random walk perturbed at extrema. Probab. Theory Relat. Fields 113(4): 501–518 · Zbl 0930.60041 [5] Feller W. (1971). An introduction to probability theory and its applications, vol. II. Wiley, New York · Zbl 0219.60003 [6] Kesten H., Kozlov M.V. and Spitzer F. (1975). A limit law for random walk in a random environment. Compositio Math. 30: 145–168 · Zbl 0388.60069 [7] Kozma, G.: Excited random walk in three dimensions has positive speed, 2003. Preprint. Available via http://arxiv.org/abs/math.PR/0310305 [8] Kozma, G.: Excited random walk in two dimensions has linear speed, 2005. Preprint. Available via http://arxiv.org/abs/math.PR/0512535 [9] 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) · Zbl 1115.60103 [10] Norris J.R. (1998). Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2. Cambridge University Press, Cambridge, Reprint of 1997 original · Zbl 0873.60043 [11] Vatutin V.A. and Zubkov A.M. (1993). Branching processes. II. J. Soviet Math. 67(6): 3407–3485, Probability theory and mathematical statistics, 1 · Zbl 0846.60083 [12] 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) [13] Zerner M.P.W. (2005). Multi-excited random walks on integers. Probab. Theory Relat. Fields 133(1): 98–122 · Zbl 1076.60088 [14] 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) · Zbl 1112.60086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.