Quenched limits for the fluctuations of transient random walks in random environment on \(\mathbb{Z}\). (English) Zbl 1279.60126
This paper deals with nearest-neighbour random walks in random environment (RWRE) on \(\mathbb{Z}\), a model introduced by F. Solomon [Ann. Probab. 3, 1–31 (1975; Zbl 0305.60029)]. The random environment is a sequence of independent and identically distributed random variables taking values in \((0,1)\), indexed by \(\mathbb{Z}\); at each step the walk goes to the left or to the right with probability depending on the value of the environment at its current location.
The authors consider transient RWRE under the quenched law, that is for a typical and fixed environment, which is relevant for applications like DNA unzipping. They are interested in the fluctuations of the hitting time of a level \(x\) around its (quenched) mean. They prove that for large \(x\) and for a set of environments arbitrarily close to 1, these fluctuations can be described explicitly by a function of the environment, and that their limiting law can be characterized by means of a Poisson point process with explicit intensity. Therefore, their result can be seen as the quenched counterpart of the one by H. Kesten, M. V. Kozlov and F. Spitzer [Compos. Math. 30, 145–168 (1975; Zbl 0388.60069)]. The proof relies on a decomposition of the so-called potential into valleys. The fluctuations of the hitting times are then related to the crossing times of valleys, the main contribution coming from deep valleys, which are shown to be asymptotically independent.
To improve readability, the result is followed by a sketch of the proof and preparatory lemmas, while some of the most technical parts are deferred to an appendix. This paper will be of interest for anyone working in the field of random walks in random environment.
The authors consider transient RWRE under the quenched law, that is for a typical and fixed environment, which is relevant for applications like DNA unzipping. They are interested in the fluctuations of the hitting time of a level \(x\) around its (quenched) mean. They prove that for large \(x\) and for a set of environments arbitrarily close to 1, these fluctuations can be described explicitly by a function of the environment, and that their limiting law can be characterized by means of a Poisson point process with explicit intensity. Therefore, their result can be seen as the quenched counterpart of the one by H. Kesten, M. V. Kozlov and F. Spitzer [Compos. Math. 30, 145–168 (1975; Zbl 0388.60069)]. The proof relies on a decomposition of the so-called potential into valleys. The fluctuations of the hitting times are then related to the crossing times of valleys, the main contribution coming from deep valleys, which are shown to be asymptotically independent.
To improve readability, the result is followed by a sketch of the proof and preparatory lemmas, while some of the most technical parts are deferred to an appendix. This paper will be of interest for anyone working in the field of random walks in random environment.
Reviewer: Julien Poisat (Leiden)
MSC:
| 60K37 | Processes in random environments |
| 60F05 | Central limit and other weak theorems |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60E10 | Characteristic functions; other transforms |
Keywords:
random walk in random environment; quenched distribution; Poisson point process; fluctuation theory for random walks; beta distributionsReferences:
| [1] | Alili, S. (1999). Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 334-349. · Zbl 0946.60046 · doi:10.1239/jap/1032374457 |
| [2] | Baldazzi, V., Cocco, S., Marinari, E. and Monasson, R. (2006). Inference of DNA sequences from mechanical unzipping: An ideal-case study. Phys. Rev. Lett. 96 128102. |
| [3] | Chamayou, J.-F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Probab. 4 3-36. · Zbl 0728.60012 · doi:10.1007/BF01046992 |
| [4] | Dolgopyat, D. and Goldsheid, I. Y. (2010+). Quenched limit theorems for nearest neighbour random walks in 1D random environment. Available at . 1012.2503 · Zbl 1260.60187 · doi:10.1007/s00220-012-1539-3 |
| [5] | Durrett, R. (2004). Probability : Theory and Examples , 3rd ed. Duxbury Press, Belmont, CA. · Zbl 1202.60001 |
| [6] | Enriquez, N., Sabot, C., Tournier, L. and Zindy, O. (2010). Stable fluctuations for ballistic random walks in random environment on \(\mathbb{Z}\). Available at . 1004.1333 · Zbl 1279.60126 |
| [7] | Enriquez, N., Sabot, C. and Zindy, O. (2009). A probabilistic representation of constants in Kesten’s renewal theorem. Probab. Theory Related Fields 144 581-613. · Zbl 1168.60034 · doi:10.1007/s00440-008-0155-9 |
| [8] | Enriquez, N., Sabot, C. and Zindy, O. (2009). Limit laws for transient random walks in random environment on \(\mathbb{Z}\). Ann. Inst. Fourier ( Grenoble ) 59 2469-2508. · Zbl 1200.60093 · doi:10.5802/aif.2497 |
| [9] | Enriquez, N., Sabot, C. and Zindy, O. (2009). Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime. Bull. Soc. Math. France 137 423-452. · Zbl 1186.60108 |
| [10] | Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166. · Zbl 0724.60076 · doi:10.1214/aoap/1177005985 |
| [11] | Goldsheid, I. Y. (2007). Simple transient random walks in one-dimensional random environment: The central limit theorem. Probab. Theory Related Fields 139 41-64. · Zbl 1134.60065 · doi:10.1007/s00440-006-0038-x |
| [12] | Iglehart, D. L. (1972). Extreme values in the \(GI/G/1\) queue. Ann. Math. Statist. 43 627-635. · Zbl 0238.60072 · doi:10.1214/aoms/1177692642 |
| [13] | Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248. · Zbl 0291.60029 · doi:10.1007/BF02392040 |
| [14] | Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compos. Math. 30 145-168. · Zbl 0388.60069 |
| [15] | Peterson, J. (2009). Quenched limits for transient, ballistic, sub-Gaussian one-dimensional random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 45 685-709. · Zbl 1178.60071 · doi:10.1214/08-AIHP149 |
| [16] | Peterson, J. and Samorodnitsky, G. (2010+). Weak quenched limiting distributions for transient one-dimensional random walk in random environment. Available at . 1011.6366 · Zbl 1277.60188 |
| [17] | Peterson, J. and Zeitouni, O. (2009). Quenched limits for transient, zero speed one-dimensional random walk in random environment. Ann. Probab. 37 143-188. · Zbl 1179.60070 · doi:10.1214/08-AOP399 |
| [18] | Resnick, S. I. (2007). Heavy-Tail Phenomena : Probabilistic and Statistical Modeling . Springer, New York. · Zbl 1152.62029 · doi:10.1007/978-0-387-45024-7 |
| [19] | Sinai, Y. G. (1982). The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 256-268. · Zbl 0505.60086 · doi:10.1137/1127028 |
| [20] | Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1-31. · Zbl 0305.60029 · doi:10.1214/aop/1176996444 |
| [21] | 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 · doi:10.1007/978-3-540-39874-5_2 |
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