The Double Dyson Index β Effect in Non-Hermitian Tridiagonal Matrices

Area of Science:

• Random Matrix Theory
• Mathematical Physics
• Symmetry in Quantum Systems

Background:

• The Dyson index (β) classifies random matrix ensembles based on symmetries.
• Standard values (1, 2, 4) correspond to orthogonal, unitary, and symplectic classes.
• In β-ensembles, β can be any positive real number, losing its role as a variable count.

Purpose of the Study:

• To investigate if the Dyson index (β) can regain its function in non-Hermitian random matrix theory.
• To explore the behavior of non-Hermitian matrices derived from real matrices with a given β.

Main Methods:

• Removing the Hermitian condition from real matrices generated with a specific β value.
• Analyzing the asymptotic behavior of the resulting non-Hermitian matrices.
• Examining three tridiagonal ensembles: β-Hermite, β-Laguerre, and β-Jacobi.

Main Results:

• Non-Hermitian matrices, with doubled independent variables, asymptotically behave as if generated with a 2β value.
• This effect restores the operative nature of the β index in these ensembles.
• The phenomenon is observed across the β-Hermite, β-Laguerre, and β-Jacobi tridiagonal ensembles.

Conclusions:

• The Dyson index (β) can be effectively restored in non-Hermitian random matrix theory.
• Removing the Hermitian condition provides a pathway to re-establish the significance of β.
• This finding has implications for understanding symmetries and variable counts in generalized random matrix ensembles.