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

TL;DR

The paper develops a random-matrix ensemble for the stationary covariance matrices of reversible multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, constrained by the Sylvester-Lyapunov relation. By applying the replica method, it derives analytic expressions for the spectra of the equal-time covariance $\boldsymbol{S}$ and the lagged covariance $\boldsymbol{S}'(\tau)$, and identifies a stability transition from a finite, positive-support spectrum to a regime with negative eigenvalues; at marginal stability the spectrum exhibits a universal tail $\rho_S(\lambda) \propto \lambda^{-5/2}$. The framework is explored for homogeneous, bimodal, and uniform temperature distributions, yielding phase diagrams and confirming universal features across distributions. This work provides a structurally informed null model for empirical covariances in complex systems and offers a route to study how temperature heterogeneity and interactions shape spectral properties of covariances and lagged covariances.

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.

Figures (6)

• Figure 1: Spectral density of the precision matrix $\boldsymbol{S}^{-1}$ (Eq. (\ref{['djfg']})) and of the covariance matrix $\boldsymbol{S}$ for stationary MVOU processes interacting through Eq. (\ref{['uyq']}) with $\alpha=1$. The temperatures are equal to $T_i=1 \,\, \forall i$. In the stable regime ($\mu > 2$), the spectral density $\rho_{S}(\lambda)$ is given by Eq. (\ref{['hyte']}), while it exhibits a power-law tail at marginal stability ($\mu=2$).
• Figure 2: Spectral density $\rho_{S^{\prime}(\tau)}(\lambda)$ of the lagged covariance matrix $\boldsymbol{S}^{\prime}(\tau)$ for stationary MVOU processes described by Eq. (\ref{['MVOU11']}). The coupling matrix is given by Eq. (\ref{['uyq1']}), with $\alpha=1$ and a homogeneous temperature $T_i=1 \,\, \forall i$. The left panel shows $\rho_{S^{\prime}(\tau)}(\lambda)$ in the stable regime ($\mu=3$), while the right panel illustrates the power-law decay of $\rho_{S^{\prime}(\tau)}(\lambda)$ at marginal stability ($\mu=2$). The solid lines are obtained from Eq. (\ref{['ytda']}), while the symbols are numerical diagonalization results derived from an ensemble of $10$ matrices $\boldsymbol{S}^{\prime}(\tau)$ with $N= 10^{4}$.
• Figure 3: (a) Spectral density $\rho_{S}(\lambda)$ of the covariance matrix $\boldsymbol{S}$ for stationary MVOU processes interacting through Eq. (\ref{['uyq']}) with $\mu=3$ and different values of $\alpha$. (b) Spectral density $\rho_{S^{\prime}(\tau)}(\lambda)$ of the lagged covariance matrix $S^{\prime}(\tau)$ for stationary MVOU processes (see Eq. (\ref{['MVOU11']})) interacting through Eq. (\ref{['uyq1']}) with $\alpha=1$ and $\mu=3$. The temperatures in both panels follow a bimodal distribution with $T_0=\delta=1$ and $p=1/2$ (see Eq. (\ref{['dist_bimod']})). The solid lines are obtained from the solutions of Eqs. (\ref{['order_param']}) and (\ref{['order_param1']}) with $\epsilon=10^{-3}$, while the symbols are numerical diagonalization results derived from an ensemble of $10$ covariance matrices with $N = 10^4$.
• Figure 4: Main panel: stability diagram $(\mu,p)$ of stationary MVOU processes with $\alpha=1$ (see Eq. (\ref{['uyq']})). The temperatures are drawn from the bimodal distribution of Eq. (\ref{['dist_bimod']}) with $T_0=1$. The MVOU process is stable above the solid lines, unstable below them, and marginally stable at the lines. The inset displays the dynamics of $m(t)$, Eq. (\ref{['mt']}), for $p=1/2$, $\delta=1$, and different $\mu$. These results follow from the numerical integration of Eq. (\ref{['MVOU']}) for an ensemble of $N=10^3$ dynamical variables. Lower panels: spectral densities of the covariance and precision matrices, $\boldsymbol{S}$ and $\boldsymbol{S}^{-1}$, across the stability transition for $T_0=\delta=1$ and $p=1/2$. These results are obtained from the solutions of Eq. (\ref{['order_param']}) with $\epsilon=10^{-6}$.
• Figure 5: (a) Spectral density $\rho_{S}(\lambda)$ of the covariance matrix $\boldsymbol{S}$ for stationary MVOU processes interacting through Eq. (\ref{['uyq']}) with $\mu=3$ and different $\alpha$. (b) Spectral density $\rho_{S^{\prime}(\tau)}(\lambda)$ of the lagged covariance matrix $S^{\prime}(\tau)$ for stationary MVOU processes (see Eq. (\ref{['MVOU11']})) interacting through Eq. (\ref{['uyq1']}) with $\alpha=1$ and $\mu=3$. The temperatures in both panels follow an uniform distribution with $T_M=3/2$ and $\Delta=1$ (see Eq. (\ref{['uni11']})). The solid lines are derived from the solutions of Eqs. (\ref{['order_param']}) and (\ref{['order_param1']}) with $\epsilon=10^{-3}$, while the symbols are numerical diagonalization results obtained from an ensemble of $10$ covariance matrices with $N = 10^4$.