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Additional resources for Statistics of energy levels and eigenfunctions in disordered systems
As was shown by the author  (and then reproduced in  within the ballistic -model approach), the far asymptotic behavior is of log-normal form and is thus in agreement with AKL. The saddle-point method of Muzykantskii and Khmelnitskii allowed also to study the asymptotic behavior of distribution functions of other quantities: relaxation times [29}31], eigenfunction intensities [32,33], local density of states , inverse participation ratio [35,36], level curvatures , etc. The form of the saddle-point solution describes directly the spatial shape of the corresponding anomalously localized state [29,36].
Usion approximation. We will see however that the saddle-point action depends on l through ln(R/l) only, and thus does not depend on A in the leading order, so that we can set A"1. 36) by (r )"0, H where the parameter r will be speci"ed below. \ eF P " ; r 4r4R . # (r /R)! 2 H The function (r) is meant as being constant within the vicinity "r"4r of the disk center. The H condition 4l\ yields r &lC. It is important to note that the result does not depend on details H of the cut-o! procedure.
70), provided the condition <" (0)"