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Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light
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Fabrication and Characterization of Disordered Polymer Optical Fibers for Transverse Anderson Localization of Light

Published on: July 29, 2013

Localization transition in symmetric random matrices.

F L Metz1, I Neri, D Bollé

  • 1Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

This study uses the cavity method to analyze random matrices, revealing a critical line for localization transitions in Lévy matrices and comparing theoretical findings with simulations.

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Area of Science:

  • Physics
  • Mathematics
  • Statistical Mechanics

Background:

  • Understanding the behavior of random matrices is crucial in various scientific fields.
  • Localization transitions in disordered systems are a key area of research.

Purpose of the Study:

  • To investigate the inverse participation ratio and localization transitions in infinite random matrices.
  • To analyze two specific ensembles: Laplacian matrices on sparse random graphs and fully connected Lévy matrices.

Main Methods:

  • The cavity method is employed to study the theoretical behavior.
  • Diagonalization of finite random matrices is used for comparison.

Main Results:

  • The study derives a critical line that distinguishes localized from extended states in Lévy matrices.
  • Behavior of the inverse participation ratio is analyzed for both matrix types.

Conclusions:

  • The cavity method provides accurate predictions for localization phenomena in random matrices.
  • Theoretical results align well with numerical simulations on finite matrices.