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Large Deviation Function for the Number of Eigenvalues of Sparse Random Graphs Inside an Interval.

Fernando L Metz1, Isaac Pérez Castillo2

  • 1Departamento de Física, Universidade Federal de Santa Maria, 97105-900 Santa Maria, Brazil.

Physical Review Letters
|September 17, 2016
PubMed
Summary
This summary is machine-generated.

We developed a method to calculate eigenvalue distributions in sparse random matrices, finding asymmetric rate functions in Erdös-Rényi graphs and the Anderson model. This asymmetry reflects distinct eigenvalue statistics and disorder effects.

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Area of Science:

  • Mathematics
  • Physics
  • Computer Science

Background:

  • Understanding eigenvalue statistics in random matrices is crucial for various scientific fields.
  • Sparse random matrices and disordered systems present unique analytical challenges.

Purpose of the Study:

  • To develop a general method for calculating the exact rate function of large deviation probabilities for eigenvalues in sparse random matrices.
  • To apply this method to analyze eigenvalue statistics in Erdös-Rényi graphs and the Anderson model on random graphs.

Main Methods:

  • A novel general method to derive the exact rate function Ψ_{[a,b]}(k) for eigenvalue distributions.
  • Application of the method to analyze the shifted index number of eigenvalues in Erdös-Rényi graphs.
  • Application to the Anderson model on regular random graphs, focusing on eigenvalue counts within spectral regions.

Main Results:

  • The derived rate function Ψ_{[a,b]}(k) is asymmetric for both Erdös-Rényi graphs and the Anderson model, unlike rotationally invariant matrices.
  • The asymmetry is linked to disorder and distinguishes localized from delocalized eigenstates.
  • Level compressibility ratio κ_{2}/κ_{1} for the Anderson model on regular graphs is found to be between 0 and 1 in the bulk regime.

Conclusions:

  • The developed method provides exact control over large deviation eigenvalue probabilities in sparse random matrices.
  • The observed asymmetry in the rate function offers new insights into the behavior of disordered systems.
  • Findings are validated through comparison with numerical diagonalization, showing good agreement.