Eigenvalue repulsion and eigenvector localization in sparse non-Hermitian random matrices.

Area of Science:

• Complex Systems
• Statistical Mechanics
• Condensed Matter Physics

Background:

• Complex networks with probabilistic, directed interactions are common in nature.
• Sparse non-Hermitian random matrices model nonequilibrium dynamics in these networks.
• Understanding eigenvalue and eigenvector behavior is crucial for network analysis.

Purpose of the Study:

• Investigate eigenvalue correlations in sparse non-Hermitian random matrices.
• Analyze eigenvector localization in one-dimensional (1D) network structures.
• Examine the impact of disordered self-interactions and directional bias.

Main Methods:

• Studied 1D spatial structures using sparse non-Hermitian random matrices.
• Applied methods from statistical mechanics of 2D interacting particles.
• Analyzed two-point eigenvalue correlations in the complex plane.

Main Results:

• Eigenvalue repulsion in the complex plane correlates with eigenvector delocalization.
• Self-interaction disorder localizes eigenvectors more than hopping disorder.
• Large cycles and directional bias resist eigenvector localization and separate eigenvectors.

Conclusions:

• Disordered non-Hermitian networks exhibit complex eigenvalue-eigenvector relationships.
• Periodic boundary conditions and directional bias offer robustness against localization.
• Findings have implications for asymmetric random networks and require new analytical tools.