Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices

Area of Science:

• Mathematics
• Physics
• Random Matrix Theory

Background:

• The Wigner semicircle distribution and Wigner surmise describe properties of random matrices.
• Standard random matrix ensembles (GOE, GUE) are Hermitian and have specific eigenvalue distributions.
• Investigating non-Hermitian matrices with inner products offers insights into generalized random matrix theory.

Purpose of the Study:

• To analyze the level density and level-spacing distribution of random matrices defined with an inner product.
• To determine if these properties generalize known results from Hermitian random matrices.
• To explore new classes of random matrices beyond standard ensembles.

Main Methods:

• Considered random matrices M = AF, where A is a real symmetric or complex Hermitian matrix and F is a diagonal inner product.
• Analyzed the self-adjoint property of M with respect to F, ensuring real eigenvalues.
• Derived the level density σ(x) and level-spacing distribution p(s) for these matrices.

Main Results:

• The level density σ(x) is independent of the entry distribution q(x) and depends only on the inner product F.
• This level density generalizes the Wigner semicircle distribution.
• The level-spacing distributions p(s) are independent of q(x) but depend on F and the nature of A (real or complex), generalizing the Wigner surmise.

Conclusions:

• Introduced F-dependent generalizations of the Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble.
• Demonstrated that inner products significantly influence random matrix properties.
• Opened avenues for studying new random matrix ensembles with tailored properties.