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

We consider quenched and annealed Lyapunov exponents for the Green's function of $-Δ+γV$, where the potentials $V(x), x\in\Z^d$, are i.i.d. nonnegative random variables and $γ>0$ is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like $c\sqrtγ$ as $γ$ 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.

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