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Edge universality for non-Hermitian random matrices.

Giorgio Cipolloni1, László Erdős1, Dominik Schröder2

  • 1IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.

Probability Theory and Related Fields
|March 12, 2021
PubMed
Summary
This summary is machine-generated.

Large non-Hermitian random matrices exhibit universal eigenvalue statistics near the unit circle. This finding mirrors the Tracy-Widom distribution observed in Wigner ensembles, extending universality to non-Hermitian systems.

Keywords:
Circular lawGinibre ensembleGirko’s formulaUniversality

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Area of Science:

  • Random Matrix Theory
  • Mathematical Physics
  • Spectral Analysis

Background:

  • Non-Hermitian random matrices are crucial in various fields, including quantum mechanics and finance.
  • Understanding their spectral properties, especially near the unit circle (spectral edge), is a key challenge.
  • Existing universality results primarily focus on Hermitian matrices (Wigner ensembles).

Purpose of the Study:

  • To investigate the local eigenvalue statistics of large non-Hermitian random matrices.
  • To determine if universality, similar to Hermitian counterparts, exists in non-Hermitian systems.
  • To establish a connection between general non-Hermitian matrices and the established Ginibre ensemble.

Main Methods:

  • Consideration of large non-Hermitian real or complex random matrices.
  • Analysis of matrices with independent, identically distributed, centered entries.
  • Focus on eigenvalue statistics in the vicinity of the unit circle (spectral edge).

Main Results:

  • The local eigenvalue statistics near the unit circle for these matrices coincide with those of the Ginibre ensemble.
  • This universality is demonstrated for matrices with independent, identically distributed, centered entries, not necessarily Gaussian.
  • The findings establish the non-Hermitian analogue of the Tracy-Widom distribution's universality.

Conclusions:

  • Universality of eigenvalue statistics extends to the spectral edges of non-Hermitian random matrices.
  • The Ginibre ensemble serves as a universal model for non-Hermitian eigenvalue distributions near the spectral edge.
  • This work bridges a gap in understanding random matrix theory by providing non-Hermitian universality results.