Fractal Structure of Spectra of Random Matrices

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Numerical investigations of the spectra of a class of non-Hermitian random matrices arising from the study of biological networks show an unexpected fractal structure, Phys. Rev. E 93, 042310. (2016). These systems are a particular type of neural networks. The resulting matrices are band and sparse matrices. It is highly unusual that spectra of random matrices show a fractal nature. Usually, eigenvalues of non-hermitian random matrices concentrate on a given domain in the complex plane and tend to have a flat density. Another interesting feature of such matrices is that for certain values of the parameters the eigenvectors are localized. All these properties suggest a rich and complicated mathematical structure that branches in different fields of mathematics, probability and physics: random matrices, biological and neural networks, Anderson localization and random graphs. At present all we know comes from numerical experiments. We propose a thorough analytical investigation of these systems. The main aims of this project are to prove that such spectra are indeed fractal in the limit of large dimension and to understand the connection and the implications on the localization properties of the eigenvectors. It would also shed light on the phenomenon of Anderson localization for Hermitian operators. The interdisciplinary nature of this project means that we will need to use techniques from different areas of mathematics: hermitation techniques and other tools from the theory of non-hermitian random matrices as well from fractal dynamical systems, asymptotic analysis and probability theory. The student will greatly benefit from learning techniques from all these different areas of mathematics. Because little is known about these systems the potential impacts outside academia are long terms, but may include artificial intelligence, applications to data science as well as medical research, as neural networks are ubiquitous in applied mathematics and engineering This project falls within the EPSRC Statistics and Applied Probability research area as well as EPSRC Mathematical Physics research area.