Statistical properties of the Green function in finite size for Anderson localization models with multifractal eigenvectors

Abstract

For Anderson Localization models with multifractal eigenvectors on disordered samples containing N sites, we analyze in a unified framework the consequences for the statistical properties of the Green function. We focus in particular on the imaginary part of the Green function at coinciding points G I xx (E − iη) and study the scaling with the size N of the moments of arbitrary indices q when the broadening follows the scaling η = c N δ . For the standard scaling regime δ = 1, we find in the two limits c ≪ 1 and c ≫ 1 that the moments are governed by the anomalous exponents ∆(q) of individual eigenfunctions, without the assumption of strong correlations between the weights of consecutive eigenstates at the same point. For the non-standard scaling regimes 0 < δ < 1, we obtain that the imaginary Green function follows some Fréchet distribution in the typical region, while rare events are important to obtain the scaling of the moments. We describe the application to the case of Gaussian multifractality and to the case of linear multifractality.

Robustness of delocalization to the inclusion of soft constraints in long-range random models

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Vibration Analysis of Driving-Point System with Uncertainties Using Polynomial Chaos Expansion

“…where Π N (•) is the probability density function (PDF), and N is the dimension of the square matrices of the system. As a consequence, the weight w is a random variable distributed in a Gaussian orthogonal ensemble (GOE) which obeys lognormal distribution N [27]. e authors in [27] have given the correlation between the log-normal distribution and the multifractal algorithm as follows:…”

“…For the sake of assessing the accuracy of the PCE method, a comparison MCS approach is used, with its results using 1 × 10 5 samples, serving as reference results. e errors between the estimated and reference responses are defined in equation (27), and CSD is defined in equation (28).…”

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