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Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.

Jesper R Ipsen1, Mario Kieburg1

  • 1Department of Physics, Bielefeld University, Postfach 100131, D-33501 Bielefeld, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 16, 2014
PubMed
Summary
This summary is machine-generated.

This study unifies the analysis of eigenvalue densities for products of rectangular random matrices. Rectangular matrices are shown to be statistically equivalent to square matrices, simplifying complex calculations.

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Area of Science:

  • Mathematics
  • Theoretical Physics

Background:

  • Random matrix theory is crucial for understanding complex systems.
  • Previous work focused on infinite matrix sizes and specific matrix types.

Purpose of the Study:

  • To develop a unified method for analyzing eigenvalue densities of rectangular random matrix products.
  • To extend the weak commutation relation to finite matrix sizes.
  • To establish connections between matrix ensembles and point processes.

Main Methods:

  • Developing a unified framework for arbitrary probability densities invariant under matrix multiplications.
  • Proving statistical equivalence between rectangular and square matrix products.
  • Deriving joint eigenvalue probability densities.
  • Applying results to Ginibre and Jacobi ensembles.

Main Results:

  • A weak commutation relation for random matrices at finite sizes is proven.
  • Joint probability densities for eigenvalues of matrix products are derived.
  • Complex matrix products yield determinantal point processes.
  • Real and quaternion matrix products correspond to Pfaffian point processes.

Conclusions:

  • The study provides a unified approach to random matrix products.
  • The findings link random matrix theory to determinantal and Pfaffian point processes.
  • Applications extend to quantum transport phenomena.