Distribution of exponential decay rates of localized eigenfunctions in finite quasi 1D disordered systems

Fyodorov Y.V., Mirlin A.D.

**For a quasi-ID disordered system of finite length ***L *(a thick wire or a random banded matrix) we found the explicit form of the distribution of the quantity r |V>(0)V>(£)|^{2}, **with **1^(0) **and ***xp(L) *being the values of an eigenfunction at the opposite ends of the sample. For a long sample the quantity — lnr is shown to have a normal distribution with a variance being twice as large as the mean. The latter determines the smallest Lyapunov exponent (inverse localization length).