Lyapunov exponents of Green's functions for random potentials tending to zero

Abstract

We consider quenched and annealed Lyapunov exponents for the Green's function of $-\Delta +\gamma V$, where the potentials $V(x),x\in {\mathbb{Z}}^d$, are i.i.d. nonnegative random variables and $\gamma >0$ is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like $c\sqrt{\gamma }$ as $\gamma$ tends to 0. Here the constant $c$ is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wei-Min Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.

16 pages, 3 figures. 1 figure added, very minor corrections. To appear in Probability Theory and Related Fields. The final publication is available at http://www.springerlink.com, see http://www.springerlink.com/content/p0873kv68315847x/?p=4106c52fc57743eba322052bb931e8ac&pi=21 .

References in corpus (1)

• Coincidence of Lyapunov exponents for random walks in weak random potentials

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