Coincidence of Lyapunov exponents for random walks in weak random potentials
Markus Flury
Source: Ann. Probab. Volume 36, Number 4
(2008), 1528-1583.
Abstract
We investigate the free energy of nearest-neighbor random walks on ℤd, endowed with a drift along the first axis and evolving in a nonnegative random potential given by i.i.d. random variables. Our main result concerns the ballistic regime in dimensions d≥4, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1217360978
Digital Object Identifier: doi:10.1214/00-AOP368
Mathematical Reviews number (MathSciNet): MR2435858
Zentralblatt MATH identifier: 1156.60076
References
[1] Albeverio, S. and Zhou, X. Y. (1996). A martingale approach to directed polymers in a random environment. J. Theoret. Probab. 9 171–189.
Mathematical Reviews (MathSciNet): MR1371075
Zentralblatt MATH: 0837.60069
Digital Object Identifier: doi:10.1007/BF02213739
[2] Bolthausen, E. (1989). A note on the diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529–534.
Mathematical Reviews (MathSciNet): MR1006293
Zentralblatt MATH: 0684.60013
Digital Object Identifier: doi:10.1007/BF01218584
Project Euclid: euclid.cmp/1104178982
[3] Carmona, P. and Hu, Y. (2002). On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 431–457.
Mathematical Reviews (MathSciNet): MR1939654
Zentralblatt MATH: 1015.60100
Digital Object Identifier: doi:10.1007/s004400200213
[4] Chayes, J. T. and Chayes, L. (1986). Ornstein–Zernike behavior for self-avoiding walks at all noncritical temperatures. Comm. Math. Phys. 105 221–238.
Mathematical Reviews (MathSciNet): MR849206
Digital Object Identifier: doi:10.1007/BF01211100
Project Euclid: euclid.cmp/1104115332
[5] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in a random environment: Path localization and strong disorder. Bernoulli 9 705–723.
Mathematical Reviews (MathSciNet): MR1996276
Digital Object Identifier: doi:10.3150/bj/1066223275
Project Euclid: euclid.bj/1066223275
[6] Essen, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrsch. Verw. Gebiete 9 290–308.
Mathematical Reviews (MathSciNet): MR231419
Digital Object Identifier: doi:10.1007/BF00531753
[7] Flury, M. (2007). Large deviations and phase transition for random walks in random nonnegative potentials. Stochastic Process. Appl. 117 596–612.
Mathematical Reviews (MathSciNet): MR2320951
Zentralblatt MATH: 05148875
Digital Object Identifier: doi:10.1016/j.spa.2006.09.006
[8] Greven, G. and den Hollander, F. (1992). Branching random walk in random environment: Phase transition for local and global growth rates. Probab. Theory Related Fields 91 195–249.
Mathematical Reviews (MathSciNet): MR1147615
Zentralblatt MATH: 0744.60079
Digital Object Identifier: doi:10.1007/BF01291424
[9] Ioffe, D. (1998). Ornstein–Zernike behavior and analyticity of shapes for self-avoiding walks on ℤd. Markov Process. Related Fields 4 323–350.
Mathematical Reviews (MathSciNet): MR1670027
Zentralblatt MATH: 0924.60086
[10] Imbrie, J. Z. and Spencer, T. (1988). Diffusion of directed polymers in a random environment. J. Statist. Phys. 52 609–626.
Mathematical Reviews (MathSciNet): MR968950
Zentralblatt MATH: 1084.82595
Digital Object Identifier: doi:10.1007/BF01019720
[11] Madras, M. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet): MR1197356
[12] Sznitman, A. S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1717054
Zentralblatt MATH: 0973.60003
[13] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34.
[14] Trachsler, M. (1999). Phase transitions and fluctuations for random walks with drift in random potentials. Ph.D. thesis, Univ. Zurich.
[15] Zerner, M. P. W. (1998). Directional decay of the Green’s function for a random nonnegative potential on ℤd. Ann. Appl. Probab. 8 246–280.
