Finding rules where randomness prevails

Assistant Professor Kevin Schnelli will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm.
Random matrix theory has developed hugely in the last decade, gaining more and more applications in statistics, financial mathematics, genetic research, technology and many other areas. The purpose of Kevin Schnelli’s project is to find universal properties that are displayed in large random matrices.

Matrices, which are rectangular arrays filled with numbers, are vital to the mathematical description of nature. Each matrix has a set of numbers, called eigenvalues, that make the matrix easier to handle.

When describing a large complex system, it is often impossible to determine the numerical values of all the elements in the matrix. Instead, they are replaced by random variables; the values are randomly selected according to a rule, the law for random variables. This approach was advocated by the famous physicist Eugene Wigner in the 1950s, when he was studying energy levels in heavy atomic nuclei. He thus started a new research area – random matrix theory.

When the elements in the matrix are chosen at random, the eigenvalues also become random. The distribution of the random eigenvalues, and how they fluctuate, is then of great importance. For example, it has been found that when the size of the matrix goes to infinity, the fluctuations in the sums of eigenvalues’ functions follow a normal distribution, that of the famous Gaussian bell curve. This is called a central limit theorem.

The aim of the proposed research is to derive similar central limit theorems for eigenvalues in multimatrix models for random matrices. The models are partly motivated by the random matrix theory and probability theory, and partly by concrete applications in physics and statistics.