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Random matrix theory reveals how correlations impact complex systems. This study provides a formula for the leading eigenvalue, showing non-diagonal correlations significantly affect stability in dynamical systems.

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Area of Science:

  • Physics
  • Mathematics
  • Complex Systems

Background:

  • Random matrix theory (RMT) analyzes eigenvalue spectra from statistical properties.
  • RMT offers insights into the stability of complex dynamical systems.
  • Understanding eigenvalue spectra is crucial for predicting system behavior.

Purpose of the Study:

  • To investigate the eigenvalue spectrum of random matrices with generalized correlations.
  • To develop an analytical method for mapping matrix resolvents to dynamical system response functions.
  • To derive a closed-form expression for the leading eigenvalue in correlated random matrices.

Main Methods:

  • Utilizing random matrix theory to study eigenvalue spectra.
  • Developing an analytical method connecting matrix resolvents to linear dynamical system response functions.
  • Employing path integral formalism to evaluate response functions.

Main Results:

  • A simple, closed-form expression for the leading eigenvalue of large, correlated random matrices was derived.
  • The study highlights the significant impact of non-diagonally opposite element correlations.
  • The developed analytical method successfully maps resolvents to response functions.

Conclusions:

  • Generalized correlations in random matrices substantially influence the eigenvalue spectrum.
  • Neglected correlations, particularly non-diagonal ones, play a critical role in system stability.
  • The findings provide a new tool for analyzing stability in complex dynamical systems.