Eigenvalues of random matrices with generalised correlations: a path integral approach

Abstract

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalised correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.

6+20 pages, 3+3 figures

References in corpus (17)

• Stability Criteria for Complex Ecosystems
• Random matrices: Universality of local eigenvalue statistics up to the edge
• The largest eigenvalue of rank one deformation of large Wigner matrices
• Non-Hermitean Random Matrix Theory: method of hermitization
• Introduction to Random Matrices - Theory and Practice
• Sparse random matrices: the eigenvalue spectrum revisited
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Transition to chaos in random networks with cell-type-specific connectivity
• Spectra of Sparse Random Matrices
• Dynamically evolved community size and stability of random Lotka-Volterra ecosystems
• Path integral methods for the dynamics of stochastic and disordered systems
• Matrix Games, Mixed Strategies, and Statistical Mechanics
• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Dispersal-induced instability in complex ecosystems
• Eigenvalue spectra of large correlated random matrices
• Universal hypotrochoidic law for random matrices with cyclic correlations
• Stability and dynamical properties of material flow systems on random networks

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• Feasibility and stability in large Lotka Volterra systems with interaction structure
• Eigenvalue spectral tails and localization properties of asymmetric networks
• A path integral approach to sparse non-Hermitian random matrices
• Impossible ecologies: Interaction networks and stability of coexistence in ecological communities
• Interplay between intraspecific suppression and environment in shaping biodiversity
• A class of 2x2 correlated random-matrix models with Brody spacing distribution