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

Leonardo Ferreira, Fernando Metz, Paolo Barucca
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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.
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具有异质温度的 Ornstein-Uhlenbeck 过程协方差矩阵的随机矩阵集合
我们为具有异质温度的多变量奥恩斯坦-乌伦贝克过程的静态协方差引入了一个随机矩阵模型,其中协方差受到西尔维斯特-利亚普诺夫方程的约束。利用复制法,我们计算了表征静止状态的等时协方差矩阵的谱密度,证明该模型经历了稳定与不稳定状态之间的转换。在稳定状态下,频谱密度具有有限的正支持,而在不稳定状态下则会出现负特征值。我们确定了分隔这两种状态的批判线,并证明频谱密度在边际稳定时呈现幂律尾,其指数与温度分布无关。此外,我们还计算了滞后协方差矩阵的谱密度,它表征了原始动态变量线性变换的静止状态。我们的随机矩阵模型对于理解复杂系统研究中出现的经验相关矩阵的谱特性具有潜在的意义。
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