Abstract
Considering the behavior at infinity of the amplitude of wave functions in one-dimensional disordered systems, as effectively described by a wave transmission through an infinitely large cluster, we derive an exact expression for the decay length ${L}_{d}(E)$ of eigenstates around energy $E$. We obtain from it simple formulas for ${L}_{d}$ at $E=0$ and in the limit of weak and strong disorder, and verify them in the case of Lorentzian type of disorder where exact results are independently available. The empirical result of Bush concerning the behavior of the maximum ${L}_{d}$ in random binary alloys is found to be approximately correct, but only in the limit of weak disorder.
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