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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
Summary
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.
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.
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