Open Conference Systems, DDAYS LAC 2024 Main Conference

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Stability and criticality of Ornstein-Uhlenbeck processes with heterogeneous temperatures via random matrix theory
Leonardo dos Santos Ferreira, Fernando Metz, Paolo Barucca

Building: Cero Infinito
Room: 1401
Date: 2024-12-13 02:40 PM – 03:00 PM
Last modified: 2024-11-22

Abstract


The application of random matrix theory (RMT) to the investigation of complex systems has shown how the spectral properties of eigenvalues and eigenvectors are efficient probes for stability and universality. With examples ranging from quantum mechanics [1] to financial data [2], the central idea of the RMT approach is to give away the search for the exact form of the microscopic nature of many-body systems, encoded by the interaction matrix, introducing instead probability distributions for its elements. Once the adequate ensemble of random matrices is chosen, based on symmetry or any other prior knowledge about the system, the task is to associate its spectral properties to the description of the desired observables.

The temporal series for the dynamical variables observed in real data are generally modeled through non-linear stochastic differential equations. Interestingly, the stability of the solutions of these equations linearized about its fixed points are crucial to the understanding of the underlying complex system. Through variation of the parameters of the model, a stable fixed point may become unstable. In the context of biology and finance [3,4], previous works have shown the fundamental role played by the empirical spectral density (ESD) of the connectivity matrix in the description of this stability transition when the linearized differential equation is no longer stochastic, i.e. in the absence of thermal fluctuations around the fixed point.

To address the effects of the stochastic fluctuations on the stability of the complex system, it is necessary to maintain the stochastic term in the linearized equation [5]. In this context, the stochastic fluctuations promote non-trivial correlations between the dynamical variables. Being an experimentally accessible quantity, the covariance matrix of the dynamical variables is a good alternative for the connectivity matrix, which is hard to recover. However, the behavior of its spectral properties across the stability transition remains elusive.

Building on previous studies [6], in this work [7] we propose a RMT ensemble for the connectivity matrix of multivariate Ornstein-Uhlenbeck (MVOU) processes, the simplest  class of linear stochastic models. We consider that the temperatures of the individual stochastic variables are heterogeneous, drawn from an arbitrary distribution. Imposing a reversibility condition, we obtain explicitly the dependence of the correlation matrix on the connectivity and diffusion matrices, that respectively control microscopic interactions and thermal fluctuations. This explicit relation stands in contrast to the majority of the previous works that rely on ad hoc models for the covariance matrix (e.g. [8]). By means of the replica method, we obtain the ESD of the stationary correlation matrix of the MVOU process, and use it to characterize the stable and unstable phases for some specific choices of temperature distributions. At the critical line that separates both regimes, the ESD of the correlation matrix presents a power law with an exponent that is independent of the temperature distribution, suggesting universal critical behavior.

References:

[1] A.D.Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems”, Physics Reports 326, 259–382(2000).

[2] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, and H.E. Stanley, “Universal and nonuniversal properties of cross correlations in financial time series”, Physical review letters 83, 1471(1999).

[3] R. M. May, “Will a large complex system be stable?”, Nature 238, 413–414 (1972).

[4] J. Moran and J. Bouchaud, “May’s instability in large economies”, Phys. Rev. E 100, 032307 (2019).

[5] C. Godrèche and J. Luck, “Characterizing the nonequilibrium stationary states of ornstein–uhlenbeck processes”, Journal of Physics A: Mathematical and Theoretical 52, 035002 (2019).

[6] P. Barucca, “Localization in covariance matrices of coupled heterogeneous ornstein-uhlenbeck processes”, Physical Review E 90, 062129 (2014).

[7] L. S. Ferreira, F. L. Metz and P. Barucca. “Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures”. arXiv preprint, url: https://arxiv.org/abs/2409.01262 (2024).

[8] L. Laloux, P. Cizeau, J. Bouchaud, and M. Potters. “Noise dressing of financial correlation matrices,” Phys. Rev. Lett. 83, 1467–1470 (1999).