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

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.