Perturbation approach to multifractal dimensions for certain critical random matrix ensembles

TL;DR

This paper develops a perturbation method to analyze the multifractal dimensions of eigenfunctions in critical random matrix ensembles, providing analytical formulas that align well with numerical results.

Contribution

It introduces a perturbation approach to compute multifractal dimensions across different regimes, extending analysis to models requiring modified perturbation techniques.

Findings

Analytical formulas match numerical calculations.

Perturbation series applicable in strong and weak multifractality regimes.

Leading-order terms derived for certain ensembles.

Abstract

Fractal dimensions of eigenfunctions for various critical random matrix ensembles are investigated in perturbation series in the regimes of strong and weak multifractality. In both regimes we obtain expressions similar to those of the critical banded random matrix ensemble extensively discussed in the literature. For certain ensembles, the leading-order term for weak multifractality can be calculated within standard perturbation theory. For other models such a direct approach requires modifications which are briefly discussed. Our analytical formulas are in good agreement with numerical calculations.