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Some consequences of exchangeability in random-matrix theory.
1Division of Engineering and Applied Science, Mail Stop 138-78, California Institute of Technology, Pasadena, CA 91125, USA. lecaer@mines.u-nancy.fr
Summary
This study derives explicit formulas for eigenvalue densities of random matrices using properties of exchangeable random variables. These formulas, confirmed by simulations, apply to real symmetric and Hermitian matrices, with extensions to more general ensembles.
Area of Science:
- Mathematical Physics
- Random Matrix Theory
- Asymptotic Analysis
Background:
- Understanding the spectral properties of random matrices is crucial in various scientific fields.
- Existing methods for calculating eigenvalue densities often have limitations for complex matrix ensembles.
- The distribution of random matrices depending on tr(H+H) is a significant area of study.
Purpose of the Study:
- To develop explicit expressions for asymptotic eigenvalue densities (rho(infinity)(lambda)) for random matrices.
- To investigate these densities for ensembles whose distribution depends solely on tr(H+H).
- To extend the findings to more general matrix ensembles.
Main Methods:
- Utilizing properties of infinite sequences of exchangeable random variables.
- Constructing eigenvalue densities by summing Wigner semicircles with variable radii and weights.
- Employing Monte Carlo simulations for validation.
Main Results:
- Derived explicit formulas for asymptotic eigenvalue densities rho(infinity)(lambda).
- Demonstrated the applicability to real symmetric and Hermitian matrices.
- Confirmed results through Monte Carlo simulations.
Conclusions:
- The properties of exchangeable random variables provide a direct route to calculating asymptotic eigenvalue densities.
- The derived method is effective for specific matrix ensembles and can be extended.
- This work offers a new analytical tool for random matrix theory.