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Lyapounov exponents and quenched large diviations for multidimensional random walk in random environment. (English) Zbl 0937.60095

Summary: Assign to the lattice sizes \(z\in\mathbb{Z}^d\) i.i.d. random \(2d\)-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 \((X_n)_{n\geq 0}\) 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 Lyapunov exponents known from Brownian motion in Poissonian potential or random walk in random potential. We follow a strategy similar to Sznitman.

MSC:

60K37 Processes in random environments
60F10 Large deviations
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60G50 Sums of independent random variables; random walks
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