Wir wenden die Zufallsmatrixtheorie auf den Lindblad-Superoperator L, d.h. den linearen Superoperator der Lindblad-Gleichung an und untersuchen die Verteilung und die Korrelationen der Eigenwerte von L zur Charakterisierung der Dynamik komplexer offener Quantensysteme. Zufallsmatrixensembles für L werden über Ensembles hermitescher und positiver Matrizen definiert, die alle freien Koeffizienten der Lindblad-Gleichung enthalten. Wir bestimmen Mittelwert und Breiten der Verteilung der von Null verschiedenen Eigenwerte von L in der komplexen Ebene und zeigen, wie diese Verteilung von den Verteilungen und Korrelationen der Eigenwerte der Koeffizientenmatrizen abhängt. In vielerlei Hinsicht ähneln die Ensembles für L dem Ginibreschen orthogonalen Ensemble. Beispielsweise finden wir das gleiche Abstoßungsverhalten zwischen benachbarten Eigenwerten. Alle Ergebnisse werden mit denen einer früheren Zufallsmatrixanalyse von Ratengleichungen verglichen. / Random matrix theory is applied to the Lindblad superoperator L, i.e., the linear superoperator of the Lindblad equation. We study the distribution and correlations of eigenvalues of L to characterize the dynamics of complex open quantum systems. Random matrix ensembles for L are given in terms of ensembles of hermitian and positive matrices, which contain all free coefficients of the Lindblad equation. We determine mean and widths of the distribution of the nonzero eigenvalues of L in the complex plane and show how this distribution depends on the distributions and correlations of eigenvalues of the matrices of coefficients. In many respects the ensembles for L resemble the Ginibre orthogonal ensemble. For instance, we find the same repulsion characteristics for neighboring eigenvalues. All results are compared to an earlier work on random matrix theory for rate equations.

Random matrix theory (RMT) is a mathematical framework that has found profound applications in physics, particularly in the study of many-body systems. Its success lies in its ability to predict universal statistical properties of complex systems, independent of the specific details. This thesis explores the application of RMT to two classes of many-body systems: quantum and stochastic many-body systems. Within the quantum framework, this work focuses on the Bose-Hubbard system, which is paradigmatic for modeling ultracold atoms in optical traps. According to RMT and the Eigenstate Thermalization Hypothesis (ETH), eigenstate-to-eigenstate fluctuations of expectation values of local observables decay rapidly with the system size in the thermodynamic limit at sufficiently large temperatures. Here, we study these fluctuations in the classical limit of fixed lattice size and increasing boson number. We find that the fluctuations follow the RMT prediction for large system sizes but deviate substantially for small lattices. Partly motivated by these results, the Bose-Hubbard model on three sites is studied in more detail. On few sites, the Bose-Hubbard model is known to be a mixed system, being neither fully chaotic nor integrable. We compare energy-resolved classical and quantum measures of chaos, which show a strong agreement. Deviations from RMT predictions are attributed to the mixed nature of the few-site model. In the context of stochastic systems, generators of Markov processes are studied. The focus is on the spectrum. We present results from two investigations of Markov spectra. First, we investigate the effect of sparsity on the spectrum of random generators. Dense random matrices previously used as a model for generic generators led to very large spectral gaps and therefore to unphysically short relaxation times. In this work, a model of random generators with adjustable sparsity — number of zero matrix elements — is presented, extending the dense framework. It is shown that sparsity leads to longer, more physically realistic relaxation times. Second, the generator spectrum of the Asymmetric Simple Exclusion Process (ASEP), a quintessential model in non-equilibrium statistical mechanics, is analyzed. We investigate the spectral boundary, which is characterized by pronounced spikes. The emergence of these spikes is analyzed from several points of view, including RMT. The results presented in this thesis contribute to the understanding of the applicability of RMT to many-body systems. This thesis highlights successes such as the explanation of “ETH fluctuations” in Bose-Hubbard models, the improvement of random matrix descriptions by introducing sparsity, and the emergence of spikes in the spectral boundary of the ASEP. The latter is a notable case where RMT provides insights even though the ASEP is a Bethe-integrable system. Furthermore, this thesis shows examples of the limits of RMT, exemplified by the results presented for the Bose-Hubbard model with a few sites.