Investigating finite-size effects in random matrices by counting resonances

TL;DR

This paper reevaluates the concept of resonances in random matrices, aiming to connect resonance counting with measurable quantities to improve finite-size analysis of Anderson localization.

Contribution

It redefines resonances in random matrices and establishes a link to measurable physical quantities, enabling finite-size system analysis.

Findings

Resonance counting can be related to measurable physical observables.

The work provides a foundation for applying resonance methods to finite systems.

It clarifies the limitations of resonance counting in the thermodynamic limit.

Abstract

Resonance counting is an intuitive and widely used tool in Random Matrix Theory and Anderson Localization. Its undoubted advantage is its simplicity: in principle, it is easily applicable to any random matrix ensemble. On the downside, the notion of resonance is ill-defined, and the `number of resonances' does not have a direct mapping to any commonly used physical observable like the participation entropy, the fractal dimensions, or the gap ratios (r-parameter), restricting the method's predictive power to the thermodynamic limit only where it can be used for locating the Anderson localization transition. In this work, we reevaluate the notion of resonances and relate it to measurable quantities, building a foundation for the future application of the method to finite-size systems. To access the HTML version of the paper & discuss it with the authors, visit https://enabla.com/pub/558.