The Annals of Probability

Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments

Noam Berger, Moran Cohen, and Ron Rosenthal

Full-text: Open access

Abstract

In this work, we discuss certain ballistic random walks in random environments on $\mathbb{Z}^{d}$, and prove the equivalence between the static and dynamic points of view in dimension $d\geq4$. Using this equivalence, we also prove a version of a local limit theorem which relates the local behavior of the quenched and annealed measures of the random walk by a prefactor.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2889-2979.

Dates
Received: August 2014
Revised: June 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139157

Digital Object Identifier
doi:10.1214/15-AOP1038

Mathematical Reviews number (MathSciNet)
MR3531683

Zentralblatt MATH identifier
1351.60132

Subjects
Primary: 60K37: Processes in random environments 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walks in random environments ballisticity equivalence of static and dynamic points of view

Citation

Berger, Noam; Cohen, Moran; Rosenthal, Ron. Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments. Ann. Probab. 44 (2016), no. 4, 2889--2979. doi:10.1214/15-AOP1038. https://projecteuclid.org/euclid.aop/1470139157


Export citation

References

  • [1] Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 334–349.
  • [2] Berger, N. (2012). Slowdown estimates for ballistic random walk in random environment. J. Eur. Math. Soc. (JEMS) 14 127–174.
  • [3] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
  • [4] Berger, N. and Deuschel, J.-D. (2014). A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment. Probab. Theory Related Fields 158 91–126.
  • [5] 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.
  • [6] Berger, N. and Zeitouni, O. (2008). A quenched invariance principle for certain ballistic random walks in i.i.d. environments. In In and Out of Equilibrium. 2. Progress in Probability 60 137–160. Birkhäuser, Basel.
  • [7] Bolthausen, E. and Sznitman, A.-S. (2002). On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 345–375.
  • [8] Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media. DMV Seminar 32. Birkhäuser, Basel.
  • [9] Campos, D. and Ramírez, A. F. (2013). Ellipticity criteria for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 160 189–251.
  • [10] Drewitz, A. and Ramírez, A. F. (2011). Ballisticity conditions for random walk in random environment. Probab. Theory Related Fields 150 61–75.
  • [11] Drewitz, A. and Ramírez, A. F. (2012). Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment. Ann. Probab. 40 459–534.
  • [12] Drewitz, A. and Ramírez, A. F. (2014). Selected topics in random walk in random environment. Topics in Percolative and Disordered Systems. Springer Proc. Math. Stat. 69 23–83.
  • [13] Guo, X. and Zeitouni, O. (2012). Quenched invariance principle for random walks in balanced random environment. Probab. Theory Related Fields 152 207–230.
  • [14] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19.
  • [15] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61–120.
  • [16] Lawler, G. F. (1982/1983). Weak convergence of a random walk in a random environment. Comm. Math. Phys. 87 81–87.
  • [17] Mathieu, P. and Piatnitski, A. (2007). Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 2287–2307.
  • [18] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 195–248. Springer, Berlin.
  • [19] 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.
  • [20] Rassoul-Agha, F. and Seppäläinen, T. (2005). An almost sure invariance principle for random walks in a space–time random environment. Probab. Theory Related Fields 133 299–314.
  • [21] Rassoul-Agha, F. and Seppäläinen, T. (2007). Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction. Ann. Probab. 35 1–31.
  • [22] 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.
  • [23] Sabot, C. (2013). Random Dirichlet environment viewed from the particle in dimension $d\geq3$. Ann. Probab. 41 722–743.
  • [24] Sidoravicius, V. and Sznitman, A.-S. (2004). Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 219–244.
  • [25] Sznitman, A.-S. (2001). On a class of transient random walks in random environment. Ann. Probab. 29 724–765.
  • [26] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509–544.
  • [27] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851–1869.
  • [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.
  • [29] Zerner, M. P. W. (2002). A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Commun. Probab. 7 191–197.