## The Annals of Probability

### On the transient (T) condition for random walk in mixing environment

Enrique Guerra Aguilar

#### Abstract

We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. Ann. Probab. 29 (2001) 724–765). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni (Israel J. Math. 148 (2005) 87–113). The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.

#### Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 3003-3054.

Dates
Received: February 2018
Revised: November 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1571731443

Digital Object Identifier
doi:10.1214/18-AOP1330

Mathematical Reviews number (MathSciNet)
MR4021243

#### Citation

Guerra Aguilar, Enrique. On the transient (T) condition for random walk in mixing environment. Ann. Probab. 47 (2019), no. 5, 3003--3054. doi:10.1214/18-AOP1330. https://projecteuclid.org/euclid.aop/1571731443

#### References

• [1] Alon, N. and Spencer, J. H. (1992). The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York.
• [2] Berger, N., Drewitz, A. and Ramírez, A. F. (2014). Effective polynomial ballisticity conditions for random walk in random environment. Comm. Pure Appl. Math. 67 1947–1973.
• [3] Chernov, A. (1962). Replication of a multicomponent chain by the lightning mechanism. Biophysics 12 336–341.
• [4] Comets, F. and Zeitouni, O. (2004). A law of large numbers for random walks in random mixing environments. Ann. Probab. 32 880–914.
• [5] Comets, F. and Zeitouni, O. (2005). Gaussian fluctuations for random walks in random mixing environments. Israel J. Math. 148 87–113.
• [6] Dobrushin, R. L. and Shlosman, S. B. (1985). Constructive criterion for the uniqueness of Gibbs field. In Statistical Physics and Dynamical Systems (Köszeg, 1984). Progress in Probability 10 347–370. Birkhäuser, Boston, MA.
• [7] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer, New York.
• [8] Drewitz, A. and Ramírez, A. F. (2014). Selected topics in random walks in random environment. In Topics in Percolative and Disordered Systems. Springer Proc. Math. Stat. 69 23–83. Springer, New York.
• [9] Guerra, E. and Ramírez, A. (2015). Almost exponential decay for the exit probability from slabs of ballistic RWRE. Electron. J. Probab. 20 no. 24, 17.
• [10] Guerra, E. and Ramírez, A. F. (2017). Asymptotic direction for random walks in mixing random environments. Electron. J. Probab. 22 Paper No. 92, 41.
• [11] Guerra, E. and Ramírez, A. F. (2018). A proof of Sznitman’s conjecture about ballistic RWRE. arXiv:1809.02011 [math.PR].
• [12] Guo, X. (2014). On the limiting velocity of random walks in mixing random environment. Ann. Inst. Henri Poincaré Probab. Stat. 50 375–402.
• [13] Iosifescu, M. and Grigorescu, Ş. (1990). Dependence with Complete Connections and Its Applications. Cambridge Tracts in Mathematics 96. Cambridge Univ. Press, Cambridge.
• [14] Kalikow, S. A. (1981). Generalized random walk in a random environment. Ann. Probab. 9 753–768.
• [15] Kesten, H. (1977). A renewal theorem for random walk in a random environment. 67–77.
• [16] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120, 238.
• [17] Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin models. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 93–191. Springer, Berlin.
• [18] Rassoul-Agha, F. (2003). The point of view of the particle on the law of large numbers for random walks in a mixing random environment. Ann. Probab. 31 1441–1463.
• [19] Rassoul-Agha, F. and Seppäläinen, T. (2009). Almost sure functional central limit theorem for ballistic random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 45 373–420.
• [20] Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
• [21] Sinaĭ, Ya. G. (1982). The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatn. Primen. 27 247–258.
• [22] Sznitman, A.-S. (2000). Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS) 2 93–143.
• [23] Sznitman, A.-S. (2001). On a class of transient random walks in random environment. Ann. Probab. 29 724–765.
• [24] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509–544.
• [25] Sznitman, A.-S. (2003). On new examples of ballistic random walks in random environment. Ann. Probab. 31 285–322.
• [26] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869.
• [27] Williams, D. (1991). Probability with Martingales. Cambridge Mathematical Textbooks. Cambridge Univ. Press, Cambridge.
• [28] Zeitouni, O. (2004). Random walks in random environment. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1837 189–312. Springer, Berlin.