{"title":"具有异质温度的 Ornstein-Uhlenbeck 过程协方差矩阵的随机矩阵集合","authors":"Leonardo Ferreira, Fernando Metz, Paolo Barucca","doi":"arxiv-2409.01262","DOIUrl":null,"url":null,"abstract":"We introduce a random matrix model for the stationary covariance of\nmultivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures,\nwhere the covariance is constrained by the Sylvester-Lyapunov equation. Using\nthe replica method, we compute the spectral density of the equal-time\ncovariance matrix characterizing the stationary states, demonstrating that this\nmodel undergoes a transition between stable and unstable states. In the stable\nregime, the spectral density has a finite and positive support, whereas\nnegative eigenvalues emerge in the unstable regime. We determine the critical\nline separating these regimes and show that the spectral density exhibits a\npower-law tail at marginal stability, with an exponent independent of the\ntemperature distribution. Additionally, we compute the spectral density of the\nlagged covariance matrix characterizing the stationary states of linear\ntransformations of the original dynamical variables. Our random-matrix model is\npotentially interesting to understand the spectral properties of empirical\ncorrelation matrices appearing in the study of complex systems.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"39 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures\",\"authors\":\"Leonardo Ferreira, Fernando Metz, Paolo Barucca\",\"doi\":\"arxiv-2409.01262\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We introduce a random matrix model for the stationary covariance of\\nmultivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures,\\nwhere the covariance is constrained by the Sylvester-Lyapunov equation. Using\\nthe replica method, we compute the spectral density of the equal-time\\ncovariance matrix characterizing the stationary states, demonstrating that this\\nmodel undergoes a transition between stable and unstable states. In the stable\\nregime, the spectral density has a finite and positive support, whereas\\nnegative eigenvalues emerge in the unstable regime. We determine the critical\\nline separating these regimes and show that the spectral density exhibits a\\npower-law tail at marginal stability, with an exponent independent of the\\ntemperature distribution. Additionally, we compute the spectral density of the\\nlagged covariance matrix characterizing the stationary states of linear\\ntransformations of the original dynamical variables. Our random-matrix model is\\npotentially interesting to understand the spectral properties of empirical\\ncorrelation matrices appearing in the study of complex systems.\",\"PeriodicalId\":501066,\"journal\":{\"name\":\"arXiv - PHYS - Disordered Systems and Neural Networks\",\"volume\":\"39 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - PHYS - Disordered Systems and Neural Networks\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.01262\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Disordered Systems and Neural Networks","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01262","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures
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.