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Random-matrix spectra as a time series.

R Fossion¹, G Torres Vargas², J C López Vieyra³

¹Instituto Nacional de Geriatría, Periférico Sur No. 2767, 10200 México D.F., Mexico and Centro de Ciencias de la Complejidad (C3), Universidad Nacional Autónoma de México, 04510 México D.F., Mexico.

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Summary

This summary is machine-generated.

This study treats random matrix eigenvalue spectra as time series, using signal analysis to differentiate trend and fluctuation modes. This method distinguishes between Poisson and Gaussian ensembles by identifying unique power-law behaviors in fluctuation modes.

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Area of Science:

* Mathematical Physics
* Statistical Mechanics
* Signal Analysis

Background:

* Random matrix theory (RMT) is crucial for understanding complex quantum systems.
* Analyzing eigenvalue spectra often involves unfolding techniques that can introduce artifacts.
* Interpreting spectra as time series offers a novel analytical perspective.

Purpose of the Study:

* To apply data-adaptive signal analysis techniques to random matrix eigenvalue spectra.
* To differentiate trend and fluctuation modes within the spectra.
* To develop an artifact-free method for distinguishing between different random matrix ensembles.

Main Methods:

* Time series analysis applied to ordered eigenvalues of finite random matrices.
* Data-adaptive decomposition into trend and fluctuation modes.
* Analysis of scale invariance and power-law behavior of fluctuation modes.

Main Results:

* Successful decomposition of spectra into distinct trend and fluctuation modes.
* Identification of scale-invariant fluctuation modes.
* Demonstration that fluctuation modes follow different power laws for Poisson and Gaussian ensembles.
* The unfolding process itself reveals ensemble-specific characteristics.

Conclusions:

* Data-adaptive signal analysis provides an effective, artifact-free method for analyzing random matrix spectra.
* The identified power-law behaviors of fluctuation modes allow for direct discrimination between Poisson and Gaussian ensembles.
* This approach offers a new perspective on the structure and properties of random matrix eigenvalues.