RanMatRanGraCircEl · Random Matrices, Random Graphs and Circular Elements

Random Matrices, Random Graphs and Circular Elements

The goal of this project is the extension of our knowledge about the structure and statistics of eigenvalues and eigenvectors of random matrices. Random matrix statistics are paradigms for collective behaviours arising when many strongly correlated random variables interact. Furthermore, random matrices frequently appear as models in applications, e.g. in natural sciences or engineering. Therefore, random matrix theory has become a very active research area at the intersection of probability theory, analysis and mathematical physics. For the last decade, random matrix theory has seen tremendous progress with major very recent developments. A large portion of this research has been devoted to rigorously understand a universality phenomenon proposed by Wigner in the 1950’s. He conjectured that the fluctuations of eigenvalue statistics emerging on the microscopic scale, i.e. on the scale of the typical eigenvalue spacing, are universal in the sense that they coincide for large classes of random matrix models and depend only on their basic symmetry type. Such universality statement can be seen as the strongly correlated analogue of the central limit theorem for weak correlations. Since the breakthroughs by Erdős, Yau and collaborators in the 2010’s, and in certain cases by Tao and Vu, universality results have been established for larger and larger classes of random matrices and a number of eigenvalue statistics. The present project aims at extending the range of validity of such universality statements to different random matrix models and observables as well as establishing and analysing the limitations of such universality phenomena. To that end, the adjacency matrices of random graphs as well as non-Hermitian random matrices are considered and the behaviour of their eigenvalues and eigenvectors is analysed. This is achieved through combining and further developing various tools from analysis, probability theory and mathematical physics. The project obtained the following significant research results. The eigenvalue density of the elliptic random matrix ensemble, a class of non-Hermitian random matrices, was understood on all mesoscopic scales. This implied complete eigenvector delocalization, a strong indication for universality. Moreover, the eigenvalue and eigenvector behaviour of Erdős–Rényi graphs was characterised in extensive detail. To that end, it was shown that, at the spectral edges, the eigenvalues form asymptotically a Poisson point process if the edge probability is sufficiently small. In a larger region near the spectral edge, each eigenvector was proved to be localized around a vertex of large degree, thus, establishing a non-universal spectral region. Furthermore, the full spectral region of delocalized eigenvectors for Erdős–Rényi graphs was determined.

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