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Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. This chapter reviews the physical origins and mathematical structures of the underlying models, and collects key predictions which give insight into the typical system behaviour. In particular, the aim is to give an idea how the different features are interlinked. The chapter mainly focuses on elastic scattering but also includes a short detour to interacting systems, which are motivated by the overarching question of ergodicity. The first sections introduce general notions from random matrix theory, such as the 10 universality classes and ensembles of Hermitian, unitary, positive-definite, and non-Hermitian matrices. The following sections then review microscopic scattering models that form the basis for statistical descriptions, and consider signatures of random scattering in decay, dynamics, and transport. The last section touches on Anderson localization and localization in interacting systems.

The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.

This chapter focuses on collective motion of isoscalar nature parameterized by shape variables. The equations of motion are derived from energy conservation as implied by self-consistency. A basic ingredient is the variation of the total static energy with deformation, which at finite thermal excitations has to be calculated for constant entropy. Linear response theory is exploited for the dynamics, especially for separating reactive and dissipative forces. Response functions for intrinsic, nucleonic motion are distinguished from those for collective dynamics. The origin of irreversible behavior due to the decay of simple to more complicated nucleonic configurations is described in detail. In practical applications, dressed single particle states are used in their dependence on temperature. The variation of the transport coefficients for inertia and friction with T obtained this way is confronted with that given in various other models, like in the diabatic one, in common RPA, in the random matrix model, or in the liquid drop model and for wall friction. Implications on rotational motion are discussed.

This chapter introduces the notion of complex scattering of waves, emphasizes the generality of the ideas involved, and explains its relevance to the field of nuclear physics and microwave cavities. It then introduces the subject of coherent wave transport through mesoscopic systems, e.g., disordered conductors and chaotic cavities, with emphasis on the statistics of fluctuations observed in these systems. These fluctuations, among them the universal conductance fluctuation, arise ultimately from the complex wave interference. Various length- and time-scales defining the mesoscopic system are discussed. The idea of maximum entropy approach (MEA) is introduced as distinct from, but related to, the idea of random-matrix theory (RMT) pioneered by Wigner originally in the context of isolated resonances of complex nuclei. The contents of this chapter include complex atomic nuclei and chaotic microwave cavities; wave localization; statistical fluctuations; mesoscopic conductors: time- and length-scales, ballistic mesoscopic cavities, diffusive mesoscopic conductors, and statistical approach to mesoscopic fluctuations.

An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.

This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.

This chapter examines nuclear properties that are related to the presence of interactions residual to the mean field. The elementary features of random matrix theory are explained, and the Wigner form of nearest level spacings in Gaussian ensembles is derived and compared with experimental evidence. Examples from nuclear physics and other chaotic and complex systems are shown. The spreading of states into more complicated configurations is described within a schematic model. For the general case, strength functions are introduced and their mathematical structure is explained. The energy dependence of single particle widths is discussed and compared with information from experiments. Inferences are drawn on the validity of the independent-particle model. For a time-dependent description, recurrent versus irreversible behavior is studied.

Data traffic in wireless networks has been increasing exponentially for a long time and is expected to continue this trend. The emerging data-hungry applications, such as video-on-demand and cloud computing, as well as the exploding number of smart user devices demand the introduction of disruptive technologies. An analogous situation appears in the case of wireline (mostly fiber-optical) traffic, where the currently deployed infrastructure is expected to soon reach its limits, leading to the so-called capacity crunch. The aim of this chapter is to introduce the physics and mathematics community to a number of relevant problems in communications research and the types of solutions that have been used to tackle them. In the process, interested readers may be able to further acquaint themselves with research in engineering bibliography cited herein.

This chapter reviews methods from random matrix theory to extract information about a large signal matrix C (for example, a correlation matrix arising in big data problems), from its noisy observation matrix M. The chapter shows that the replica method can be used to obtain both the spectral density and the overlaps between noise-corrupted eigenvectors and the true ones, for both additive and multiplicative noise. This allows one to construct optimal rotationally invariant estimators of C based on the observation of M alone. This chapter also discusses the case of rectangular correlation matrices and the problem of random singular value decomposition.

This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.