1$ (see [@Fl08], [@Zy09], [@KMZ11], [@IV12c], [@Zy12], and references therein). Lyapunov exponents represent the exponential decay rates of the quenched and annealed survival probabilities, ${P}^\omega(\tau_y<\infty)$ and ${\mathbb{E}}({P}^\omega(\tau_y<\infty))$ respectively. The equality or non-equality of quenched and annealed Lyapunov exponents determines whether the disorder introduced by the random environment is “weak” or “strong”. According to this classification, any non-trivial disorder in the one-dimensional case is strong (as well as in dimensions 2 and 3 under mild additional conditions on $V$, see Theorem 1 and a paragraph after it in [@Zy12]). Our results provide more refined information about differences between quenched and annealed behavior.

In dimensions 4 and higher one expects a transition from weak to strong disorder for i.i.d. potentials of the form $\gamma V$, $V\ge
0$, for some $\gamma^*\in(0,\infty)$. Such result is known to hold when $V$ is bounded away from $0$ ([@Fl08], [@Zy09], [@IV12c]).

Let us discuss the case of a “small” potential, i.e. the potential of the form $\gamma V$ where $\gamma\ll 1$, in more detail. We shall restrict ourselves to dimension 1 but we believe that a similar result holds in all dimensions (when the quenched speed is well defined, see Remark \[rem\] above).

Let $(V(x,\cdot))$, $x\in{\mathbb{Z}}$, be 