Eigenvalue Repulsion and Eigenfunction Localization in Sparse Non-Hermitian Random Matrices

TL;DR

This paper investigates the spectral properties of sparse non-Hermitian random matrices in 1D, revealing how eigenvalue repulsion relates to eigenvector localization and how network structure influences these phenomena.

Contribution

It introduces a novel analysis of eigenvalue correlations and localization in sparse non-Hermitian matrices, emphasizing the role of network topology and disorder.

Findings

Eigenvalue repulsion correlates with eigenvector delocalization.

Self-interaction disorder promotes eigenvector localization.

Large cycles and boundary conditions resist eigenvector localization.

Abstract

Complex networks with directed, local interactions are ubiquitous in nature, and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context, and are key to describing statistical properties of the non-equilibrium dynamics that emerges from interactions within the network structure. Here, we study one-dimensional (1d) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point eigenvalue correlations in the complex plane for sparse non-Hermitian random matrices using methods developed for the statistical mechanics of inhomogeneous 2d interacting particles. We find that eigenvalue repulsion in the complex plane directly correlates with eigenvector delocalization. In addition, for 1d chains and rings with both disordered nearest neighbor connections and self-interactions, the self-interaction disorder tends to de-correlate eigenvalues and localize eigenvectors more than simple hopping disorder. However, remarkable resistance to eigenvector localization by disorder is provided by large cycles, such as those embodied in 1d periodic boundary conditions under strong directional bias. The directional bias also spatially separates the left and right eigenvectors, leading to interesting dynamics in excitation and response. These phenomena have important implications for asymmetric random networks and highlight a need for mathematical tools to describe and understand them analytically.