An introduction to random matrix theory and locals laws

Jacob Fronk 

The field of random matrix theory (RMT) aims to investigate the properties of matrices with random entries, such as their eigenvalue or eigenvector statistics. A remarkable property of random matrices is that they obey what is known as local laws. That is, their eigenvalues converge towards a deterministic density on all mesoscopic scales as the dimension of the matrix tends to infinity. In this talk I will provide an overview of RMT and local laws, as well as their applications to physics. I will also introduce some important tools used to study them. If time permits, I will present an original result about the local law of the spherical ensemble, a specific function of non-Hermitian random matrices.