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

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

Phys Rev E

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA.

Published: November 2019

AI Article Synopsis

  • Complex networks with directed interactions often have random connections due to inherent unpredictability and disorder in their environments.
  • This study focuses on the behavior of sparse non-Hermitian random matrices within one-dimensional structures, revealing how certain interactions affect eigenvalue correlations and eigenvector localization.
  • The findings show that larger cycles in the network can help resist localization effects, and the bias in directionally influencing connections can create distinct dynamics in how the network responds and excites.

Article 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 nonequilibrium 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 two-dimensional 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 decorrelate 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.

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Source
http://dx.doi.org/10.1103/PhysRevE.100.052315DOI Listing

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