Power-law random banded matrices and ultrametric matrices: eigenvector distribution in the intermediate regime

TL;DR

This paper numerically investigates eigenvector distributions in power-law random banded and ultrametric matrices within the intermediate regime, revealing a universal distribution described by the generalized hyperbolic distribution, distinct from Porter-Thomas.

Contribution

It introduces a numerical analysis of eigenvector distributions in the intermediate regime, identifying the generalized hyperbolic distribution as a universal descriptor.

Findings

Eigenvector distributions are well described by the generalized hyperbolic distribution.

Distributions differ significantly from the Porter-Thomas distribution.

Eigenstates are extended with finite moments in the large matrix limit.

Abstract

The power-law random banded matrices and the ultrametric random matrices are investigated numerically in the regime where eigenstates are extended but all integer matrix moments remain finite in the limit of large matrix dimensions. Though in this case standard analytical tools are inapplicable, we found that in all considered cases eigenvector distributions are extremely well described by the generalised hyperbolic distribution which differs considerably from the usual Porter-Thomas distribution but shares with it certain universal properties.