A balanced excited random walk

Itaı Benjamini; Gady Kozma; Bruno Schapira

doi:10.1016/j.crma.2011.02.018

Probability Theory

A balanced excited random walk
[Une marche excité équilibrée]

Présenté par : Marc Yor

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

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 459-462.

Résumé
Abstract

Nous étudions le processus suivant sur $Z^{4}$ . 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 $Z^{4}$ 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 : 2010-09-07
Accepté le : 2011-02-21
Publié le : 2011-03-10

DOI : 10.1016/j.crma.2011.02.018

Affiliations des auteurs :

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

@article{CRMATH_2011__349_7-8_459_0,
     author = {Ita{\i} Benjamini and Gady Kozma and Bruno Schapira},
     title = {A balanced excited random walk},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {459--462},
     publisher = {Elsevier},
     volume = {349},
     number = {7-8},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.018},
     language = {en},
}

TY  - JOUR
AU  - Itaı Benjamini
AU  - Gady Kozma
AU  - Bruno Schapira
TI  - A balanced excited random walk
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 459
EP  - 462
VL  - 349
IS  - 7-8
PB  - Elsevier
DO  - 10.1016/j.crma.2011.02.018
LA  - en
ID  - CRMATH_2011__349_7-8_459_0
ER  -

%0 Journal Article
%A Itaı Benjamini
%A Gady Kozma
%A Bruno Schapira
%T A balanced excited random walk
%J Comptes Rendus. Mathématique
%D 2011
%P 459-462
%V 349
%N 7-8
%I Elsevier
%R 10.1016/j.crma.2011.02.018
%G en
%F CRMATH_2011__349_7-8_459_0

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/

Bibliographie
Cité par

[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

Cité par Sources :

Commentaires - Politique