×

Local limit theorem and equivalence of dynamic and static points of view for certain ballistic random walks in i.i.d. environments. (English) Zbl 1351.60132

Authors’ 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 \( d\geq 4\). Using this equivalence, we also prove a version of a local limit theorem which relates the local behaviour of the quenched and annealed measures of the random walk by a prefactor.

MSC:

60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 334-349. · Zbl 0946.60046
[2] Berger, N. (2012). Slowdown estimates for ballistic random walk in random environment. J. Eur. Math. Soc. ( JEMS ) 14 127-174. · Zbl 1247.60138
[3] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83-120. · Zbl 1107.60066
[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. · Zbl 1356.60175
[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. · Zbl 1364.60140
[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. · Zbl 1173.82324
[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. · Zbl 1079.60079
[8] Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media. DMV Seminar 32 . Birkhäuser, Basel. · Zbl 1075.60128
[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. · Zbl 1306.60151
[10] Drewitz, A. and Ramírez, A. F. (2011). Ballisticity conditions for random walk in random environment. Probab. Theory Related Fields 150 61-75. · Zbl 1225.60157
[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. · Zbl 1245.60099
[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. · Zbl 1329.82056
[13] Guo, X. and Zeitouni, O. (2012). Quenched invariance principle for random walks in balanced random environment. Probab. Theory Related Fields 152 207-230. · Zbl 1239.60092
[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. · Zbl 0588.60058
[15] Kozlov, S. M. (1985). The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 61-120. · Zbl 0592.60054
[16] Lawler, G. F. (1982/1983). Weak convergence of a random walk in a random environment. Comm. Math. Phys. 87 81-87. · Zbl 0502.60056
[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. · Zbl 1131.82012
[18] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 195-248. Springer, Berlin. · Zbl 0927.60027
[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. · Zbl 1039.60089
[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. · Zbl 1088.60094
[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. · Zbl 1126.60090
[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. · Zbl 1176.60087
[23] Sabot, C. (2013). Random Dirichlet environment viewed from the particle in dimension \(d\geq3\). Ann. Probab. 41 722-743. · Zbl 1269.60077
[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. · Zbl 1070.60090
[25] Sznitman, A.-S. (2001). On a class of transient random walks in random environment. Ann. Probab. 29 724-765. · Zbl 1017.60106
[26] Sznitman, A.-S. (2002). An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 509-544. · Zbl 0995.60097
[27] Sznitman, A.-S. and Zerner, M. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851-1869. · Zbl 0965.60100
[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. · Zbl 1060.60103
[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. · Zbl 1008.60107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.