Random Matrices, Random Graphs and Circular Elements

Random matrix statistics are a paradigm for the collective behaviour of many strongly correlated random variables. The proposed projects will fundamentally advance our knowledge about random matrices in novel directions. We study spectral properties of random matrices when the matrix size becomes large.

More specifically, we establish the universality of the fluctuations of the smallest singular value of almost square random matrices with independent entries.

Moreover, we determine the asymptotic eigenvalue density of non-normal random matrices with correlated entries of general expectation and the Brown measure of operator-valued circular elements.

We also obtain a central limit theorem for the difference of the linear statistics of a matrix with independent, identically distributed entries and its minor. Furthermore, we analyse the spectra of random graphs.

Specifically, a transition in the eigenvalue fluctuations of very sparse Erdos-Renyi graphs, the eigenvector delocalisation of directed Erdos-Renyi graphs as well as the extreme eigenvalues and eigenvectors of preferential attachment graphs. Finally, we investigate a variational problem motivated by wireless communication.

The techniques proposed for these projects comprise a variety of tools from analysis (spectral theory, variational methods), probability theory (stochastic differential equations, large deviation bounds) and mathematical physics (self-consistent equations). For the purpose of these projects, the tools mentioned above will be developed further.