The Annals of Probability

Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment

Martin P. W. Zerner

Full-text: Open access

Abstract

Assign to the lattice sizes $z \epsilon \mathbb{Z}^d$ i.i.d. random 2 $d$-dimensional vectors $(\omega(z, z + e))_{|e|=1}$ whose entries take values in the open unit interval and add up to one. Given a realization $\omega$ of this environment, let $(X_n)_{n \geq o}$ be a Markov chain on $\mathbb{Z}^d$ which, when at $z$, moves one step to its neighbor $z + e$ with transition probability $\omega(z, z + e)$. We derive a large deviation principle for $X_n/n$ by means of a result similar to the shape theorem of first-passage percolation and related models. This result produces certain constants that are the analogue of the Lyapounov exponents known from Brownian motion in Poissonian potential or random walk in random potential. We follow a strategy similar to Sznitman.

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1446-1476.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855870

Digital Object Identifier
doi:10.1214/aop/1022855870

Mathematical Reviews number (MathSciNet)
MR1675027

Zentralblatt MATH identifier
0937.60095

Subjects
Primary: 60F10: Large deviations 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random walk in random environment shape theorem Lyapounov exponent large deviations

Citation

Zerner, Martin P. W. Lyapounov exponents and quenched large deviations for multidimensional random walk in random environment. Ann. Probab. 26 (1998), no. 4, 1446--1476. doi:10.1214/aop/1022855870. https://projecteuclid.org/euclid.aop/1022855870


Export citation

References

  • 1 DEMBO, A., PERES, Y. and ZEITOUNI, O. 1996. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 667 683.
  • 2 DEMBO, A. and ZEITOUNI, O. 1993. Large Deviations Techniques. Jones and Bartlett, Boston.
  • 3 FREIDLIN, M. 1985. Functional Integration and Partial Differential Equations. Princeton Univ. Press.
  • 4 GANTERT, N. and ZEITOUNI, O. 1998. Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 177 190.
  • 5 GREVEN, A. and DEN HOLLANDER, F. 1994. Large deviations for a random walk in random environment. Ann. Probab. 22 1381 1428.
  • 6 HUGHES, B. D. 1996. Random Walks and Random Environments 2. Clarendon Press, Oxford.
  • 7 KALIKOW, S. A. 1981. Generalized random walk in a random environment. Ann. Probab. 9 753 768.
  • 8 KESTEN, H. 1986. Aspects of first passage percolation. Ecole d'ete de Probabilites de St. ´ ´ ´ Flour. Lecture Notes in Math. 1180 125 264. Springer, Berlin.
  • 9 KESTEN, H. 1993. On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296 338.
  • 10 LEE, T.-Y. and TORCASO, F. 1998. Wave propagation in a lattice KPP equation in random media. Ann. Probab. 26 1179 1197.
  • 11 LIGGETT, T. 1985. Interacting Particle Systems. Springer, New York.
  • 12 LORENTZEN, L. and WAADELAND, H. 1992. Continued Fractions with Applications. NorthHolland, Amsterdam.
  • 13 PETROV, V. V. 1995. Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford.
  • 14 PISZTORA, A. and POVEL, T. 1997. Large deviation principle for random walk in a quenched random environment in the low speed regime. Unpublished manuscript.
  • 15 PISZTORA, A., POVEL, T. and ZEITOUNI, O. 1997. Precise large deviation estimates for one-dimensional random walk in random environment. Probab. Theory Related Fields. To appear.
  • 16 REVESZ, P. 1990. Random Walk in Random and Non-Random Environments. World ´ ´ Scientific, Singapore.
  • 17 SOLOMON, F. 1975. Random walks in random environment. Ann. Probab. 3 1 31.
  • 18 SZNITMAN, A. S. 1994. Shape theorem, Lyapounov exponents, and large deviations for Brownian motion in a Poissonian potential. Comm. Pure Appl. Math. 47 1655 1688.
  • 19 SZNITMAN, A. S. 1996. Distance fluctuations and Lyapounov exponents. Ann. Probab. 24 1507 1530.
  • 20 SZNITMAN, A. S. 1998. Brownian Motion, Obstacles and Random Media. Springer, Berlin.
  • 21 ZERNER, M. P. W. 1998. Directional decay of the Green's function for a random nonnegative potential on d. Ann. Appl. Probab. 8 246 280.