Non-Hermitian random matrices: theory and applications

Random Matrix Theory (RMT) serves already for some time as a paradigmatic framework for successfully describing so-called "universal" properties of physical systems, both quantum and classical, identifying those features which are largely independent of particular detail of underlying structures. The universal characteristics then serve as fingerprints for particular classes of systems of very different microscopic nature. For long time the main effort in RMT was concentrated on studies of Hermitian matrices, one of the reasons being that the Hermiticity condition in Quantum Mechanics is necessary to ensure conservation of the number of particles. Similarly, in Classical Mechanics, and eventually in Statistical Mechanics based on it, a somewhat analogous property is known as "reciprocity of interactions" (at the lowest level encapsulated, in particular, in obeying Newton’s third law "action equals reaction"). In the last decade it however became clear that systems with non-Hermitian (respectively, non-reciprocal) interactions do appear in Nature, and show many properties which are not seen in their Hermitian counterparts. In classical systems of many interacting degrees of freedom nonreciprocal interactions are operative under far-from-equilibrium conditions, in particular they play a crucial role in numerous biological (or "active matter") processes, including neural networks dynamics. In quantum systems, non-Hermiticity appears as a result of taking into account various gain and loss mechanisms due to ever-present environment, and is especially important in systems with many-body quantum chaos. All these developments increased interest in analysis of properties of non-Hermitian random matrices of various types. Among other characteristics, properties of eigenvectors of such matrices are very different from their Hermitian counterparts and remain poorly studied, despite some progress achieved in recent years. Bringing our understanding of eigenvectors of non-Hermitian random matrices at higher level poses a serious challenge, as addressing their properties is highly nontrivial and requires new insights and techniques. The present project aims to study in a systematic way those and other related objects for several classes of non-Hermitian random matrices, in particular arising in applications in physics, and beyond. This knowledge is important, among other things, for understanding mechanisms controlling stability of non-Hermitian and non-reciprocal systems, as well as for more detailed description of scattering of waves in disordered media. Any progress in such studies is expected to provide solid mathematical basis for reliably identifying some universal characteristics of behaviour of systems with violated Hermiticity or/and presence of non-reciprocal interactions.