A law of large numbers for random walks in random mixing environments.(English)Zbl 1078.60089
Summary: We prove a law of large numbers for a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments of some random times depending on the path. It applies in the nonnestling case, but we also provide examples of nestling walks that satisfy our assumptions. The derivation is based on an adaptation, using coupling, of the regeneration argument of Sznitman and Zerner.
MSC:
| 60K37 | Processes in random environments |
| 60F15 | Strong limit theorems |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
Cite
References:
| [1] | Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximations . Oxford Univ. Press. ·Zbl 0746.60002 |
| [2] | Berbee, H. (1987). Convergence rates in the strong law for a bounded mixing sequence. Probab. Theory Related Fields 74 253–270. ·Zbl 0587.60028 ·doi:10.1007/BF00569992 |
| [3] | Bradley, R. (1989). A caution on mixing conditions for random fields. Statist. Probab. Lett. 8 489–491. ·Zbl 0697.60054 ·doi:10.1016/0167-7152(89)90032-1 |
| [4] | Dobrushin, R. and Shlosman, S. (1985). Constructive criterion for the uniqueness of Gibbs fields. In Statistical Physics and Dynamical Systems (J. Fritz, A. Jaffe and D. Szász, eds.) 347–370. Birkhäuser, Basel. ·Zbl 0569.46042 |
| [5] | Doukhan, P. (1994). Mixing : Properties and Examples . Springer, New York. ·Zbl 0801.60027 |
| [6] | Föllmer, H. (1988). Random fields and diffusion processes. École d’Été de Probabilités de Saint-Flour 1985–1987 . Lecture Notes in Math. 1362 101–203. Springer, Berlin. ·Zbl 0661.60063 |
| [7] | Kalikow, S. A. (1981). Generalized random walks in random environment. Ann. Probab. 9 753–768. JSTOR: ·Zbl 0545.60065 |
| [8] | Komorowski, T. and Krupa, G. (2001). Random walk in a random environement with correlated sites. J. Appl. Probab. 38 1018–1032. ·Zbl 1003.60094 ·doi:10.1239/jap/1011994189 |
| [9] | Komorowski, T. and Krupa, G. (2002). The law of large numbers for ballistic, multi-dimensional random walks on random lattices with correlated sites. Ann. Inst. H. Poincaré Probab. Statist. 39 263–285. ·Zbl 1017.60105 ·doi:10.1016/S0246-0203(02)00002-X |
| [10] | Kozlov, S. M. (1985). The method of averaging and walks in inhomogeneous environments. Russian Math. Surveys 40 73–145. ·Zbl 0615.60063 ·doi:10.1070/RM1985v040n02ABEH003558 |
| [11] | Martinelli, F. (1999). Lectures on Glauber dynamics for discrete spin systems. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1717 96–181. Springer, Berlin. ·Zbl 1051.82514 |
| [12] | Molchanov, S. A. (1994). Lectures on random media. Lectures on Probability Theory. Lecture Notes in Math. 1581 242–411. Springer, Berlin. ·Zbl 0814.60093 |
| [13] | 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 ·doi:10.1214/aop/1055425786 |
| [14] | Shen, L. (2002). A law of large numbers and a central limit theorem for biased random motions in random environment. Ann. Appl. Probab. 12 477–510. ·Zbl 1016.60092 |
| [15] | Sznitman, A. S. (2000). Slowdown estimates and central limit theorem for random walks in random environment. JEMS 2 93–143. ·Zbl 0976.60097 ·doi:10.1007/s100970000017 |
| [16] | Sznitman, A. S. (2001). On a class of transient random walks in random environment. Ann. Probab. 29 724–765. ·Zbl 1017.60106 ·doi:10.1214/aop/1008956691 |
| [17] | Sznitman, A. S. (2002). Lectures on random motions in random media. In Ten Lectures on Random Media . Birkhäuser, Basel. ·Zbl 1075.60128 |
| [18] | 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 ·doi:10.1214/aop/1022874818 |
| [19] | Thorisson, H. (2000). Coupling , Stationarity , and Regeneration . Springer, New York. ·Zbl 0949.60007 |
| [20] | Williams, D. (1991). Probability with Martingales . Cambridge Univ. Press. ·Zbl 0722.60001 ·doi:10.1017/CBO9780511813658 |
| [21] | Zeitouni, O. (2004). Lecture notes on random walks in random environment. Lecture Notes in Math. Springer, New York. ·Zbl 1060.60103 |
| [22] | Zerner, M. P. W. and Merkl, F. (2001). A zero–one law for planar random walks in random environment. Ann. Probab. 29 1716–1732. ·Zbl 1016.60093 ·doi:10.1214/aop/1015345769 |
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