Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Persistence of some additive functionals of Sinai’s walk

Alexis Devulder

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We are interested in Sinai’s walk $(S_{n})_{n\in\mathbb{N}}$. We prove that the annealed probability that $\sum_{k=0}^{n}f(S_{k})$ is strictly positive for all $n\in[1,N]$ is equal to $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, for a large class of functions $f$, and in particular for $f(x)=x$. The persistence exponent $\frac{3-\sqrt{5}}{2}$ first appears in a nonrigorous paper of Le Doussal, Monthus and Fischer, with motivations coming from physics. The proof relies on techniques of localization for Sinai’s walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a two-sided Brownian motion.


Nous nous intéressons à la marche de Sinai $(S_{n})_{n\in\mathbb{N}}$. Nous prouvons que la probabilité annealed que $\sum_{k=0}^{n}f(S_{k})$ soit strictement positive pour tout $n\in[1,N]$ est égale à $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, pour une large classe de fonctions $f$, et en particulier pour $f(x)=x$. L’exposant de persistance $\frac{3-\sqrt{5}}{2}$ est d’abord apparu dans un article non rigoureux de Le Doussal, Monthus et Fischer, avec des motivations venant de la physique. La preuve est basée sur des techniques de localisation pour la marche de Sinai et utilise des résultats de Cheliotis sur les changements de signe des fonds de vallées d’un mouvement Brownien indexé par $\mathbb{R}$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1076-1105.

Received: 19 February 2014
Revised: 10 March 2015
Accepted: 7 April 2015
First available in Project Euclid: 28 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60J55: Local time and additive functionals

Random walk in random environment Sinai’s walk Integrated random walk One-sided exit problem Persistence Survival exponent


Devulder, Alexis. Persistence of some additive functionals of Sinai’s walk. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1076--1105. doi:10.1214/15-AIHP679.

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  • [1] P. Andreoletti. Localisation et Concentration de la Marche de Sinai. Ph.D. thesis, Université Aix-Marseille II, 2003. Available at
  • [2] P. Andreoletti. Alternative proof for the localization of Sinai’s walk. J. Stat. Phys. 118 (2005) 883–933.
  • [3] P. Andreoletti and A. Devulder. Localization and number of visited valleys for a transient diffusion in random environment. Electron. J. Probab. 20 (56) (2015) 1–58.
  • [4] F. Aurzada and T. Simon. Persistence probabilities & exponents. Preprint, 2012. Available at arXiv:1203.6554.
  • [5] A. Bovier and A. Faggionato. Spectral analysis of Sinai’s walk for small eigenvalues. Ann. Probab. 36 (2008) 198–254.
  • [6] A. J. Bray, S. N. Majumdar and G. Schehr. Persistence and first-passage properties in non-equilibrium systems. Advances in Physics 62 (2013) 225–361.
  • [7] T. Brox. A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14 (1986) 1206–1218.
  • [8] D. Cheliotis. Diffusion in random environment and the renewal theorem. Ann. Probab. 33 (2005) 1760–1781.
  • [9] D. Cheliotis. Localization of favorite points for diffusion in a random environment. Stochastic Process. Appl. 118 (2008) 1159–1189.
  • [10] S. Cocco and R. Monasson. Reconstructing a random potential from its random walks. Europhysics Letters 81 (2008) 20002.
  • [11] A. Dembo, J. Ding and F. Gao. Persistence of iterated partial sums. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 873–884.
  • [12] A. Devulder. Some properties of the rate function of quenched large deviations for random walk in random environment. Markov Process. Related Fields 12 (2006) 27–42.
  • [13] A. Devulder. The speed of a branching system of random walks in random environment. Statist. Probab. Lett. 77 (2007) 1712–1721.
  • [14] N. Enriquez, C. Lucas and F. Simenhaus. The arcsine law as the limit of the internal DLA cluster generated by Sinai’s walk. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 991–1000.
  • [15] A. O. Golosov. Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 (1984) 491–506.
  • [16] P. Le Doussal, C. Monthus and D. Fisher. Random walkers in one-dimensional random environments; exact renormalization group analysis. Phys. Rev. E 59 (1999) 4795–4840.
  • [17] Y. Hu. Tightness of localization and return time in random environment. Stochastic Process. Appl. 86 (2000) 81–101.
  • [18] B. D. Hughes. Random Walks and Random Environment, Vol. II: Random Environments. Oxford Science Publications, Oxford, 1996.
  • [19] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent RV’s and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131.
  • [20] J. Neveu and J. Pitman. Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. In Séminaire de Probabilités XXIII 239–247. Lecture Notes in Math. 1372. Springer, Berlin, 1989.
  • [21] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific, Singapore, 2005.
  • [22] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 2nd edition. Springer, Berlin, 1994.
  • [23] S. Schumacher. Diffusions with random coefficients. Contemp. Math. 41 (1985) 351–356.
  • [24] Z. Shi. Sinai’s walk via stochastic calculus. In Milieux aléatoires 53–74. Panor. Synthèses 12. Soc. Math. France, Paris, 2001.
  • [25] T. Simon. The lower tail problem for homogeneous functionals of stable processes with no negative jumps. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 165–179.
  • [26] Ya. G. Sinai. The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 (1982) 256–268.
  • [27] Ya. G. Sinai. Distribution of some functionals of the integral of a random walk. Theoret. and Math. Phys. 90 (1992) 219–241.
  • [28] F. Solomon. Random walks in a random environment. Ann. Probab. 3 (1975) 1–31.
  • [29] H. Tanaka. Localization of a diffusion process in a one-dimensional Brownian environment. Comm. Pure Appl. Math. 47 (1994) 755–766.
  • [30] V. Vysotsky. On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178–1193.
  • [31] O. Zeitouni. Lectures notes on random walks in random environment. In Lectures on Probability Theory and Statistics 193–312. Lecture Notes Math. 1837. Springer, Berlin, 2004.
  • [32] O. Zindy. Upper limits of Sinai’s walk in random scenery. Stochastic Process. Appl. 118 (2008) 981–1003.