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Characteristic polynomials of random matrices at edge singularities

Brezin1, Hikami

  • 1Laboratoire de Physique Theorique de l'Ecole Normale Superieure, Unite Mixte de Recherche 8549 du Centre National de la Recherche Scientifique et de l'Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris Cedex 05, France.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
Summary

This study explores universal properties of random matrix theory, focusing on characteristic polynomial moments at the spectrum edge and with external sources. These findings reveal simplified formulas for complex random matrix behaviors.

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Area of Science:

  • Random Matrix Theory
  • Mathematical Physics

Background:

  • Correlation functions of random variables det(lambda-X) are key in random matrix theory.
  • Moments of these distributions exhibit universality at bulk spectrum levels.

Purpose of the Study:

  • Investigate modified universality classes at the spectrum edge (Airy/Bessel problems).
  • Analyze phenomena arising from adding external matrix sources to probability distributions.
  • Derive simple formulas for computing characteristic polynomial moments in these modified scenarios.

Main Methods:

  • Analysis of correlation functions for random matrices.
  • Examination of spectral edge behavior and crossover phenomena.
  • Inclusion of external matrix sources and tuning to critical points.

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Main Results:

  • Identified universal functions for moments at the spectrum edge.
  • Described crossover to Airy or Bessel problems.
  • Derived simple, general formulas for moments with arbitrary external sources.

Conclusions:

  • Extended understanding of universality in random matrix theory to spectrum edges and external source effects.
  • Demonstrated the existence of simple formulas for complex random matrix distributions.
  • Provided a framework for computing characteristic polynomial moments in diverse random matrix ensembles.