Random Matrix Theory with Applications

Dear Colleagues,

In 1951, Wigner proposed the random matrix theory (RMT) as a procedure to describe universal statistical properties of the energy levels of atomic nuclei and of complex atoms and molecules. In his approach, the Hamiltonian of the system is replaced by an ensemble of random Hermitian matrices with Gaussian distributed elements belonging to the corresponding universality class. Shortly after, Dyson proposed the three circular ensembles of random unitary matrices, and French the embedded ensembles. Furthermore, the Rosenzweig–Porter ensemble of random matrices interpolating between collective and complex motion in the nuclear many-body system, and the Ginibre ensembles of non-Hermitian random matrices were then introduced. The statistical fluctuations of scattering processes were also first investigated in the context of compound nuclear reactions. In the Heidelberg approach, the scattering matrix describing such reactions is expressed in terms of the nuclear Hamiltonian which is replaced by a random matrix from the Gaussian ensembles. The derivation of analytical expressions for universal statistical properties of the eigenvalues of random Hermitian and non-Hermitian matrices or of the scattering-matrix elements within the Heidelberg approach was significantly advanced by the introduction of Efetov’s supersymmetry technique and the ensuing mapping of the random-matrix problem onto a non-linear supersymmetric σ model. In 1984, Bohigas, Gianonni and Schmit applied the Wigner’s RMT approach to the eigenenergies of a single-particle quantum system whose classical dynamics were fully chaotic. This led to the formulation of the Bohigas–Gianonni–Schmit conjecture which triggered an enormous amount of theoretical and experimental investigations and the development of random-matrix models beyond the Gaussian ensembles. Starting from the early 1990s, the RMT approach was applied in lattice Quantum Chromodynamics to describe the spectral properties of the Wilson–Dirac operator or low-energy excitations in disordered superconductors. This led to an extension of Dyson’s threefold approach consisting of three symmetry classes to the tenfold approach. In recent years, there has been a huge advance in the extension of these RMT models within the research field of many-body quantum chaos. These models are typically applied to static Hamiltonians. Other approaches focus on dynamical aspects, which, in fact, were already included at an early stage of RMT in Dyson’s Brownian motion model for random matrices that evolve stochastically over time.

The aim of the Special Issue is to provide an overview of ongoing RMT research in the context of nuclear physics, quantum chaotic scattering, recent developments in the framework of single- and many-body quantum chaos and progress in the mathematical approach.

Prof. Dr. Barbara Dietz
Guest Editor

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• random matrix theory