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 -Delta + gamma V, where the potentials V(x) ,x epsilon 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 root 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 Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.