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Summing free unitary random matrices.

Andrzej Jarosz¹

¹Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Kraków, Poland. jedrekjarosz@gmail.com

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

|August 27, 2011

Summary

This study introduces quaternion free probability calculus to analyze sums of random matrices. It derives eigenvalue and singular value densities for large matrices and proves a central limit theorem.

Area of Science:

Mathematics
Quantum Physics
Random Matrix Theory

Background:

Free probability theory is extended to non-Hermitian matrices using quaternion free probability calculus.
Understanding the behavior of sums of independent unitary random matrices is crucial in various fields.

Purpose of the Study:

To derive the mean densities of eigenvalues and singular values for sums of independent unitary random matrices in the large-size limit.
To solve and numerically test "master equations" for specific cases of circular unitary ensemble summands.
To prove a central limit theorem and its correction for independent, identically distributed, zero-drift unitary random matrices.

Main Methods:

Application of quaternion free probability calculus to non-Hermitian matrices.
Derivation of large-size limit properties for sums of independent unitary random matrices.

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Solving and numerical testing of "master equations" for specific cases.

Conjecturing and proving central limit theorems for random matrices.

Main Results:

The mean densities of eigenvalues and singular values of sums of independent unitary random matrices are derived in the large-size limit.
Two "master equations" are formulated and solved for circular unitary ensemble summands, with numerical verification.
A finite-size extension of the results is conjectured using the complementary error function.
A central limit theorem and its first subleading correction are proven for independent, identically distributed, zero-drift unitary random matrices.

Conclusions:

Quaternion free probability calculus provides a powerful framework for analyzing sums of random matrices.
The derived densities and theorems offer new insights into the spectral properties of random matrix sums.
The study bridges theoretical advancements with numerical verification and conjectures for future research.