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Correlation Properties of Sparse Real Symmetric Random Matrix

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Part of the NATO ASI Series book series (ASIC, volume 200)

Abstract

Sparse real symmetric random matrices are studied because they model quantum mechanics problems like coupled oscillators. By numerical calculations we determine the correlation properties of sets of eigenvalues like Nearest Neighbor Distribution, NND and spectral rigidity. We deal with the transition from Poisson statistics to Gaussian Orthogonal Ensemble (GOE) statistics as a function normalized “size” of off diagonal matrix elements. The NND are fitted with one parameter, Brody like, distributions. The transitions of NND from Poisson to Wigner is governed by the normalized mean of absolute value of off diagonal matrix elements. The long range correlation properties are examined through the Fourier transform of the stick spectrum of eigenvalues. Statistical analysis of eigenvectors shows that, for intermediate coupling, i.e. between Poisson and GOE, each eigenvector is a superposition of Lo equally weighted (statistically) basis vectors. Lo is given by a dimensionless Fermi golden rule.

Keywords

Diagonal Element Random Matrix Coupling Parameter Correlation Property Couple Oscillator 
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Copyright information

© D. Reidel Publishing Company 1987

Authors and Affiliations

  • R. Jost
    • 1
  1. 1.Service National des Champs Intenses CNRSGrenoble CedexFrance

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