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

TL;DR

This paper introduces a random matrix model for the covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, analyzing spectral properties and stability transitions using the replica method.

Contribution

It provides a novel random matrix framework for stationary covariance of heterogeneous Ornstein-Uhlenbeck processes, including spectral density calculations and stability analysis.

Findings

Spectral density exhibits a finite support in stable regimes.

Negative eigenvalues appear in unstable regimes.

Power-law tail at marginal stability with temperature-independent exponent.

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.