Crossover in nonstandard random-matrix spectral fluctuations without unfolding

Area of Science:

• Mathematical Physics
• Random Matrix Theory
• Spectral Analysis

Background:

• Standard Gaussian ensembles in random-matrix theory have been analyzed using Singular Value Decomposition (SVD) to study spectral fluctuations.
• Traditional methods often involve unfolding procedures, which can introduce artifacts.
• SVD offers a potential alternative for analyzing spectral properties without these artifacts.

Purpose of the Study:

• To apply Singular Value Decomposition (SVD) directly to the β-Hermite ensemble and a sparse matrix ensemble.
• To decompose spectra into trend and fluctuation modes to understand spectral properties.
• To avoid artifacts associated with traditional unfolding techniques and perform data-adaptive unfolding.

Main Methods:

• Direct application of Singular Value Decomposition (SVD) to β-Hermite and sparse matrix ensembles.
• Decomposition of spectral data into trend and fluctuation modes.
• Calculation of spectral fluctuation measures using trend modes for data-adaptive unfolding.

Main Results:

• Fluctuation modes exhibit a crossover between soft and rigid behaviors, consistent with known results.
• SVD successfully avoids potential artifacts introduced by unfolding techniques.
• Consistent calculation of ensemble-averaged and individual-spectrum averaged statistics within a normal mode basis.

Conclusions:

• Singular Value Decomposition (SVD) provides a robust method for analyzing spectral fluctuations in random matrix ensembles.
• The approach allows for data-adaptive unfolding and accurate calculation of spectral statistics, bypassing limitations of traditional methods.
• SVD offers a consistent framework for analyzing both ensemble-averaged and individual spectral properties.