## The Annals of Probability

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

Martin P. W. Zerner

#### 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

#### 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

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