Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures

Abstract

We introduce a random matrix model for the stationary covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, where the covariance is constrained by the Sylvester-Lyapunov equation. Using the replica method, we compute the spectral density of the equal-time covariance matrix characterizing the stationary states, demonstrating that this model undergoes a transition between stable and unstable states. In the stable regime, the spectral density has a finite and positive support, whereas negative eigenvalues emerge in the unstable regime. We determine the critical line separating these regimes and show that the spectral density exhibits a power-law tail at marginal stability, with an exponent independent of the temperature distribution. Additionally, we compute the spectral density of the lagged covariance matrix characterizing the stationary states of linear transformations of the original dynamical variables. Our random-matrix model is potentially interesting to understand the spectral properties of empirical correlation matrices appearing in the study of complex systems.

arXiv comment10 pages, 6 figures

References in corpus (31)

• Stability Criteria for Complex Ecosystems
• Noise Dressing of Financial Correlation Matrices
• Universal and non-universal properties of cross-correlations in financial time series
• A Random Matrix Approach to Cross-Correlations in Financial Data
• Statistics of energy levels and eigenfunctions in disordered systems
• Colloquium: Criticality and dynamical scaling in living systems
• Correlation of financial markets in times of crisis
• Cleaning large correlation matrices: tools from random matrix theory
• Structure of Stochastic Dynamics near Fixed Points
• Signal and Noise in Correlation Matrix
• A model for correlations in stock markets
• Nonlinear Analogue of the May-Wigner Instability Transition
• Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks
• May's Instability in Large Economies
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Temporal correlations versus noise in the correlation matrix formalism: an example of the brain auditory response
• Spectral Moments of Correlated Wishart Matrices
• Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble
• Characterising the nonequilibrium stationary states of Ornstein-Uhlenbeck processes
• Linear stability analysis for large dynamical systems on directed random graphs
• A Random Matrix Approach to VARMA Processes
• Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks
• Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks
• Large deviation theory for diluted Wishart random matrices
• Inferring hidden states in Langevin dynamics on large networks: Average case performance
• Cleaning large-dimensional covariance matrices for correlated samples
• Localization in covariance matrices of coupled heterogenous Ornstein-Uhlenbeck processes
• Eigenvalue and Eigenvector Statistics in Time Series Analysis
• Inequalities for selected eigenvalues of the product of matrices
• The Spectral Boundary of Block Structured Random Matrices
• Global correlation matrix spectra of the surfacetemperature of the Oceans from Random MatrixTheory to Poisson fluctuations