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Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.

László Erdős1, Torben Krüger2, Dominik Schröder1,3

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

Communications in Mathematical Physics
|August 25, 2020
PubMed
Summary
This summary is machine-generated.

Complex Wigner-type matrices exhibit universal local eigenvalue statistics, forming a Pearcey process at cusp singularities. This resolves the final universality type for the Wigner-Dyson-Mehta conjecture in the complex Hermitian class.

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Area of Science:

  • Random Matrix Theory
  • Mathematical Physics
  • Spectral Theory

Background:

  • Wigner-type matrices are Hermitian random matrices with independent entries.
  • Eigenvalue distribution singularities, particularly cusps, were not fully understood regarding universality.
  • Previous work established universality in bulk and edge regions for these matrices.

Purpose of the Study:

  • To demonstrate universality of local eigenvalue statistics at cusp singularities in complex Wigner-type matrices.
  • To resolve the last remaining universality type of the Wigner-Dyson-Mehta conjecture for the complex Hermitian class.
  • To analyze both exact and approximate cusp singularities.

Main Methods:

  • Analysis of complex Wigner-type matrices with independent, non-identically distributed entries.
  • Proving an optimal local law at the cusp for both symmetry classes.
  • Establishing the emergence of a Pearcey process at cusp singularities.

Main Results:

  • Local eigenvalue statistics at cusp singularities are universal and follow a Pearcey process.
  • This universality holds for both exact and approximate cusps, leading to an extended Pearcey process.
  • An optimal local law at the cusp was proven, serving as a key technical ingredient.

Conclusions:

  • The study completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the complex Hermitian class.
  • The findings have implications for understanding spectral properties of non-Hermitian random matrices.
  • The established cusp universality mechanism is crucial for related research in random matrix theory.