Path-integral approach to sparse non-Hermitian random matrices

Area of Science:

• Statistical Physics
• Complex Systems Theory
• Random Matrix Theory

Background:

• Large random matrices are crucial for analyzing disordered systems across various scientific fields.
• Established laws like the elliptic law apply to dense matrices, aiding stability analysis.
• Universal statistical laws for sparse random matrices have remained elusive.

Purpose of the Study:

• To derive generalized laws for sparse random matrices, accounting for connectivity.
• To extend the applicability of random matrix theory to more complex systems.
• To develop a flexible framework for analyzing higher-order statistics in random matrices.

Main Methods:

• An expansion in inverse connectivity was performed to derive modified laws.
• A dynamical approach mapped random matrix resolvents to linear dynamical system response functions.
• Path integral formalism and Feynman diagrams were employed for perturbative analysis.

Main Results:

• General modified elliptic and semicircle laws were derived for sparse random matrices.
• A non-Hermitian generalization of the Marchenko-Pastur law was obtained.
• The framework accommodates non-Gaussian statistics in dense random matrix ensembles.

Conclusions:

• The developed dynamical and path integral methods provide a powerful tool for sparse random matrix analysis.
• This work bridges a gap in random matrix theory, enabling stability analysis in sparsely connected systems.
• The framework's versatility is demonstrated by its application to non-Hermitian matrices and higher-order statistics.