Fractal Structure of Spectra of Random Matrices

数值研究的一类非厄米特随机矩阵的光谱所产生的生物网络的研究显示出一个意想不到的分形结构，物理评论E 93，042310。（2016年）。这些系统是一种特殊类型的神经网络。得到的矩阵是带矩阵和稀疏矩阵。随机矩阵的谱具有分形性质是极不寻常的。通常，非厄米特随机矩阵的特征值集中在复平面上的一个给定区域上，并且往往具有平坦的密度。这种矩阵的另一个有趣的特征是，对于某些参数值，特征向量是局部化的。所有这些性质表明了一个丰富而复杂的数学结构，分支在不同的数学，概率和物理领域：随机矩阵，生物和神经网络，安德森本地化和随机图。目前我们所知道的一切都来自数值实验。我们建议对这些系统进行彻底的分析调查。这个项目的主要目的是证明这样的光谱确实是分形的大尺寸的限制，并了解连接和本地化的特征向量的性质的影响。这也将有助于解释厄米算符的安德森局域化现象。这个项目的跨学科性质意味着我们将需要使用来自不同数学领域的技术：Hermitation技术和其他工具，从非厄米随机矩阵理论以及分形动力系统，渐近分析和概率论。学生将极大地受益于从所有这些不同领域的数学学习技巧。由于对这些系统知之甚少，学术界以外的潜在影响是长期的，但可能包括人工智能，数据科学的应用以及医学研究，因为神经网络在应用数学和工程中无处不在。

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.