Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Martingale defocusing and transience of a self-interacting random walk

Yuval Peres, Bruno Schapira, and Perla Sousi

Full-text: Open access


Suppose that $(X,Y,Z)$ is a random walk in $\mathbb{Z}^{3}$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.


Supposons que $(X,Y,Z)$ soit une marche aléatoire dans $\mathbb{Z}^{3}$ qui se déplace de la façon suivante : à la première visite en un site, seule la coordonnée $Z$ saute de $\pm1$ avec probabilité uniforme, et aux visites suivantes en ce site $(X,Y)$ effectue un saut dans l’ensemble $\{(\pm1,0),(0,\pm1)\}$ avec probabilité uniforme. Nous montrons que cette marche est transiente, répondant ainsi à une question de Benjamini, Kozma et Schapira. Un ingrédient important de la preuve est un résultat de dispersion pour les martingales.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1009-1022.

Received: 29 April 2014
Revised: 10 December 2014
Accepted: 19 December 2014
First available in Project Euclid: 28 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Transience Martingale Self-interacting random walk Excited random walk


Peres, Yuval; Schapira, Bruno; Sousi, Perla. Martingale defocusing and transience of a self-interacting random walk. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1009--1022. doi:10.1214/14-AIHP667.

Export citation


  • [1] K. S. Alexander. Controlled random walk with a target site. Electron. Commun. Probab. 18 (2013) 43.
  • [2] O. Angel, N. Crawford and G. Kozma. Localization for linearly edge reinforced random walks. Duke Math. J. 163 (5) (2014) 889–921.
  • [3] S. N. Armstrong and O. Zeitouni. Local asymptotics for controlled martingales, 2014. Available at arXiv:1402.2402.
  • [4] A.-L. Basdevant, B. Schapira and A. Singh. Localization of a vertex reinforced random walk on $\mathbb{Z} $ with sub-linear weight. Probab. Theory Related Fields 159 (1–2) (2014) 75–115.
  • [5] R. F. Bass, X. Chen and J. Rosen. Moderate deviations for the range of planar random walks. Mem. Amer. Math. Soc. 198 (929) (2009) 1–82.
  • [6] R. F. Bass and T. Kumagai. Laws of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Probab. 30 (3) (2002) 1369–1396.
  • [7] I. Benjamini, G. Kozma and B. Schapira. A balanced excited random walk. C. R. Math. Acad. Sci. Paris 349 (7–8) (2011) 459–462.
  • [8] I. Benjamini and D. B. Wilson. Excited random walk. Electron. Commun. Probab. 8 (2003) 86–92 (electronic).
  • [9] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 2010.
  • [10] A. Erschler, B. Tóth and W. Werner. Stuck walks. Probab. Theory Related Fields 154 (1–2) (2012) 149–163.
  • [11] O. Gurel-Gurevich, Y. Peres and O. Zeitouni. Localization for controlled random walks and martingales. Electron. Commun. Probab. 19 (2014) 24.
  • [12] E. Kosygina and M. P. W. Zerner. Excited random walks: Results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N.S.) 8 (1) (2013) 105–157.
  • [13] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge, 2010.
  • [14] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI, 2009. With a chapter by Propp, James G. and Wilson, David B.
  • [15] F. Merkl and S. W. W. Rolles. Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 (5) (2009) 1679–1714.
  • [16] R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1–79.
  • [17] Y. Peres, S. Popov and P. Sousi. On recurrence and transience of self-interacting random walks. Bull. Braz. Math. Soc. (N.S.) 44 (4) (2013) 841–867.
  • [18] O. Raimond and B. Schapira. Random walks with occasionally modified transition probabilities. Illinois J. Math. 54 (4) (2012) 1213–1238. 2010.
  • [19] C. Sabot and P. Tarres. Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. 17 (2015) 2353–2378.
  • [20] P. Tarrès. Vertex-reinforced random walk on $\mathbb{Z}$ eventually gets stuck on five points. Ann. Probab. 32 (3B) (2004) 2650–2701.
  • [21] B. Tóth. Self-interacting random motions. In European Congress of Mathematics I 555–564. Barcelona, 2000. Progr. Math. 201. Birkhäuser, Basel, 2001.