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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Spectra of sparse non-hermitian random matrices: an analytical solution.

I Neri1, F L Metz

  • 1Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France.

Physical Review Letters
|August 7, 2012
PubMed
Summary

We derived an exact formula for the spectrum of sparse non-Hermitian random matrices, extending known random-matrix theories. This finding impacts transport processes on sparse graphs, revealing non-monotonic dependencies on edge symmetry.

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Area of Science:

  • * Theoretical Physics
  • * Mathematics
  • * Network Science

Background:

  • * Random-matrix theory (RMT) is crucial for understanding complex systems.
  • * Sparse non-Hermitian random matrices model various physical phenomena.
  • * Existing RMT results, like the Kesten-McKay measure and Girko's elliptic law, lack sparse generalizations.

Purpose of the Study:

  • * To derive the exact analytical expression for the spectrum of a sparse non-Hermitian random matrix ensemble.
  • * To generalize the Kesten-McKay measure and Girko's elliptic law to sparse matrices.
  • * To investigate the impact of graph symmetry on transport processes in sparse networks.

Main Methods:

  • * Exact analytical derivation of the spectral density for sparse non-Hermitian random matrices.
  • * Generalization of established RMT results (Kesten-McKay, Girko's elliptic law).
  • * Analytical study of transport processes on sparse random graphs with varying edge symmetry.

Main Results:

  • * An exact analytical expression for the spectrum of sparse non-Hermitian random matrices was obtained.
  • * This expression unifies and extends the Kesten-McKay measure and Girko's elliptic law.
  • * The convergence rate of transport processes on sparse graphs exhibits non-monotonic dependence on edge symmetry.

Conclusions:

  • * The derived spectral formula provides a powerful tool for analyzing sparse non-Hermitian systems.
  • * The study offers new insights into physical problems modeled by sparse random graphs.
  • * Non-monotonic dependence of transport rates on graph symmetry highlights complex network behaviors.