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Generation and Coherent Control of Pulsed Quantum Frequency Combs

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Published on: June 8, 2018

Quaternionic R transform and non-Hermitian random matrices.

Zdzislaw Burda¹, Artur Swiech²

¹Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, PL-30059 Kraków, Poland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 15, 2015

Summary

This summary is machine-generated.

This study rephrases non-Hermitian random matrix theory using the Cayley-Dickson construction. It introduces a quaternionic R transform to generate matrix averages and calculate eigenvalue densities.

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Area of Science:

Mathematical Physics
Quantum Field Theory
Random Matrix Theory

Background:

The study reviews the non-Hermitian diagrammatic formalism for large non-Hermitian random matrices.
This formalism generalizes free probability calculus.
The R transform is a key generating function for planar cumulants.

Purpose of the Study:

To rephrase and review the non-Hermitian diagrammatic formalism using the Cayley-Dickson construction.
To demonstrate the quaternionic R transform's ability to generate connected averages of random matrix powers.
To calculate limiting eigenvalue densities for products of Gaussian random matrices.

Main Methods:

Utilizing the Cayley-Dickson construction to generalize the free probability calculus.
Defining and applying a quaternionic extension of the R transform.
Deriving a specific R transform for Gaussian elliptic laws as a linear quaternionic map.
Calculating eigenvalue densities using the derived R transform.

Main Results:

The quaternionic R transform generates all connected averages of X and X† for large N.
The R transform for Gaussian elliptic laws is a five-parameter linear quaternionic map.
The method allows for the calculation of limiting eigenvalue densities for products of Gaussian random matrices.

Conclusions:

The Cayley-Dickson construction provides a powerful framework for non-Hermitian random matrix theory.
The quaternionic R transform is a versatile tool for analyzing random matrix properties.
This approach facilitates the computation of spectral properties for complex matrix products.