Wegner estimates and universality for non-Hermitian matrices

In the theory of probability the Central Limit Theorem (CLT) explains why the distribution of the mean of random variables from any distribution follows the Gaussian curve. Although the first observation of this behaviour is due to de Moivre in the 18th century, it was only in the 20th century that the CLT was rigorously proven. A similar phenomenon appears in the theory of random matrices and is called universality. In this context, the universality conjecture asserts that the eigenvalue statistics of large random matrices depend only on the symmetries of the matrices, and are independent of the precise probability densities that govern their stochastic behavior. Furthermore, in the limit of large dimension, the eigenvalues are distributed as if the entries were drawn from the Gaussian distribution. This conjecture is akin to the CLT and has deep philosophical and practical consequences. It is observed in numerics and experiments that many physical systems demonstrate the same behavior independently of the precise details of interactions among their constituent elements. This property of random matrices is conjectured to hold more generally. In particular, it fulfils one of the central objectives in mathematical physics: the derivation of macroscopic properties of large systems, despite unknown or random specifics of interactions. So far, it has been proven only in a few specific cases, such as the Wigner ensemble.

The goal of this project is to prove universality of various eigenvalue statistics for ensembles consisting of non-Hermitian random matrices. Such ensembles have been studied in both mathematics and physics literature and despite far-reaching applications in various fields of study, are not very well understood. By looking at the work on Hermitian random matrices, we can postulate which theorems should be true in the non-Hermitian case. Despite the slow and steady progress in the area, we see that several difficult and important problems yet remain to be solved. In particular, the study of eigenvalue correlation functions still falls far behind its Hermitian counterpart.

Non-hermitian random matrices have several applications in the technology industry, more specifically they appear in wireless communications as well in nanotechnology (growth problems). The construction and use of nanosize objects in the industry is a recent development of this century and has been coined as
nanotechnology. It suggests a whole new kind of technology based on the idea of using smaller components where quantum mechanical effects become apparent. But
the scientific knowledge and study of the quantum aspects of these objects has been developing since much earlier and was the propeller of this new field. The
expected end results set in this research proposal are essentially oriented to deepen our understanding of the spectral properties, which in turn affect the
electronic behaviour, of generic systems. The development of this kind of knowledge at the University of Bristol will enhance British excellence and
competitiveness through potential collaborations or direct dissemination of results with other groups within the University, such as the Center for
Nanoscience and Quantum Information, and in the UK. The University of Bristol has also research exchanges with local industry (research labs of Toshiba and HP
are in the Bristol area).

A direct application of the results in this proposal will be to heart models, in particular to a new technology for pacemakers to treat tachyarrythmia. This biological system is directly related to the subject matter of this grant. The object to study is 2D lattice of interacting calcium release units (CRUs) in the sinoatrial node cell and the properties of calcium waves that such a lattice supports. The amount of calcium inside the cell varies with time, so the system is an open system best described by non-Hermitian operators. The results of our proposed theoretical study will provide a much needed mathematical framework to the field of cardiac pacemaking and lead to clarity in understanding and computations. In particular the results will be applied to resolve the current controversy of heart impulse initiation (subcellular vs. cell membrane origin) in terms of basic science, and to develop new approaches to treat cardiac arrhythmia (one of the leading cause of death worldwide), in terms of applied science.