Comptes Rendus
no. 7-8
Probability Theory
A balanced excited random walk
[Une marche excité équilibrée]

Présenté par : Marc Yor

Itaı Benjamini1 ; Gady Kozma1 ; Bruno Schapira2
1 The Weizmann Institute of Science, Rehovot POB 76100, Israel
2 Département de Mathématiques, bâtiment 425, Université Paris-Sud 11, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 459-462.

Nous étudions le processus suivant sur Z4. À la première visite en un site, les deux premières coordonnées effectuent un saut dʼune marche simple (2-dimensionnelle). Aux visites suivantes en ce site, ce sont les deux dernières coordonnées qui effectuent un saut de marche simple. Nous montrons que ce processus est presque sûrement transitoire. Nous discutons également des dimensions inférieures et divers généralisations et questions connexes sont proposées.

The following random process on Z4 is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.02.018
Itaı Benjamini 1 ; Gady Kozma 1 ; Bruno Schapira 2

1 The Weizmann Institute of Science, Rehovot POB 76100, Israel
2 Département de Mathématiques, bâtiment 425, Université Paris-Sud 11, 91405 Orsay cedex, France 
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Itaı Benjamini; Gady Kozma; Bruno Schapira. A balanced excited random walk. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 459-462. doi : 10.1016/j.crma.2011.02.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.018/

[1] G. Amir; I. Benjamini; G. Kozma Excited random walk against a wall, Probab. Theory Related Fields, Volume 140 (2008), pp. 83-102

[2] I. Benjamini; D.B. Wilson Excited random walk, Electron. Commun. Probab., Volume 8 (2003), pp. 86-92 (electronic)

[3] W. Feller An Introduction to Probability Theory and Its Applications, vol. II, John Wiley & Sons, 1971

[4] M. Holmes Excited against the tide: A random walk with competing drifts (preprint) | arXiv

[5] H. Kesten; O. Raimond; Br. Schapira Random walks with occasionally modified transition probabilities (preprint) | arXiv

[6] E. Kosygina; M.P.W. Zerner Positively and negatively excited random walks on integers, with branching processes, Electron. J. Probab., Volume 13 (2008), pp. 1952-1979

[7] G. Kozma, Problem session, in: Non-classical interacting random walks, Oberwolfach report 27/2007, http://www.mfo.de/programme/schedule/2007/21/OWR_2007_27.pdf.

[8] G.F. Lawler Intersections of Random Walks, Probab. Appl., Birkhäuser Boston, Inc., Boston, MA, 1991 (219 pp)

[9] G.F. Lawler; V. Limic Random Walk: A Modern Introduction, Cambridge Stud. Adv. Math., vol. 123, Cambridge Univ. Press, Cambridge, 2010

[10] M. Menshikov; S. Popov; A. Ramirez; M. Vachkovskaia On a general many-dimensional excited random walk (preprint) | arXiv

[11] F. Merkl; S.W.W. Rolles Linearly edge-reinforced random walks, IMS Lecture Notes, Monogr. Ser. Dynamics & Stochastics, vol. 48, Inst. Math. Statist., Beachwood, OH, 2006, pp. 66-77

[12] R. Pemantle A survey of random processes with reinforcement, Probab. Surv., Volume 4 (2007), pp. 1-79 (electronic)

[13] M.P.W. Zerner Multi-excited random walks on integers, Probab. Theory Related Fields, Volume 133 (2005), pp. 98-122

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