# Wegner estimates and universality for non-Hermitian matrices

Lead Research Organisation:
University of Bristol

Department Name: Mathematics

### Abstract

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.

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.

### Planned Impact

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.

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.

### Publications

Atkin M
(2016)

*Random Matrix Ensembles with Singularities and a Hierarchy of Painlevé III Equations*in International Mathematics Research Notices
Cunden F
(2016)

*A shortcut through the Coulomb gas method for spectral linear statistics on random matrices*in Journal of Physics A: Mathematical and Theoretical
Cunden F
(2016)

*Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma*in Journal of Statistical Physics
Cunden F
(2016)

*Large- N expansion for the time-delay matrix of ballistic chaotic cavities*in Journal of Mathematical Physics
Cunden F
(2016)

*Large deviations of spread measures for Gaussian matrices*in Journal of Statistical Mechanics: Theory and Experiment
Cunden F
(2015)

*A unified fluctuation formula for one-cut ß -ensembles of random matrices*in Journal of Physics A: Mathematical and Theoretical
Cunden F
(2015)

*Joint statistics of quantum transport in chaotic cavities*in EPL (Europhysics Letters)
Cunden F
(2019)

*Moments of Random Matrices and Hypergeometric Orthogonal Polynomials*in Communications in Mathematical Physics
Cunden F
(2017)

*Density and spacings for the energy levels of quadratic Fermi operators*in Journal of Mathematical Physics
Cunden F
(2016)

*Correlators for the Wigner-Smith time-delay matrix of chaotic cavities*in Journal of Physics A: Mathematical and TheoreticalDescription | The research under this award has developed in three directions: 1) We have looked at higher order phase transitions in certain Coulomb gases, namely one-component plasma of electric charges, whose properties are markedly different from what is usually expected. Indeed, we have found that these Coulomb gases undergo a fourth order phase transition, which is weaker than that in similar systems. We also studied their large deviation distributions. 2) Electrical properties in quantum transport of microscopic cavities are a topic of intense research both experimentally and theoretically, and are of vital importance for future technologies. We discovered that the moments of the distribution of the time that an electron spends in a microscopic cavity are integer numbers that are linked to specific combinatorial objects. This has open the possibility of using tools from combinatorics to study quantum transport. 3) We studied the spectra of free fermions and quasi-free fermions. We proved the convergence of the density of states and studied the distribution of the differences of consecutive level. The Boltzmann factor of certain systems of free fermions can be expressed in terms of eigenvalue distributions of random metric ensembles. This allowed us to study the thermodynamics of such systems. We also proved universality theorems for systems of free fermions that do not correspond to random matrix ensembles. |

Exploitation Route | Part 1), 2D-component plasma of electric charges are not understood very well, and more research needs to be done in order to understand their properties, both in the mathematics and physics communities. Part 2) Combinatorics and quantum transport don't seem to have anything in common at first sight, but our research has shown that they do. This connection needs further investigation, but potentially in the future could help the design of new macroscopic devices. Part 3) The connection between random matrix theory and systems of free fermions has been a topic of intense research recently, because of its links with the KPZ (Kardar-Parisi-Zhang) equation, whose universality class apply to growth models, many interacting particle systems and polymers in random environments. Therefore, our results for free fermions have implications in many areas of solid state physics. |

Sectors | Digital/Communication/Information Technologies (including Software),Electronics,Other |