Eigenvalues and singular values of products of rectangular gaussian random matrices.

Area of Science:

• Random Matrix Theory
• Mathematical Physics

Background:

• Understanding the behavior of eigenvalues and singular values is crucial in various scientific fields.
• Gaussian random matrices are fundamental tools in statistical mechanics and quantum information.

Purpose of the Study:

• To derive exact analytic expressions for eigenvalue and singular value distributions.
• To investigate the behavior of these distributions in the limit of large matrix dimensions.
• To develop corrections for finite-sized matrices.

Main Methods:

• Derivation of exact analytic expressions.
• Analysis in the limit of large matrix dimensions.
• Development of heuristic finite size corrections.

Main Results:

• Identified power-law behavior at zero for both eigenvalue and singular value distributions.
• Determined the specific powers associated with this behavior.
• Validated heuristic finite size corrections for approximating distributions of finite matrices.

Conclusions:

• The study provides precise mathematical descriptions for random matrix products.
• The findings offer insights into the statistical properties of large random matrices.
• The developed corrections enhance the applicability of theoretical models to real-world finite systems.