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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.