Beijing Institute of Mathematical Sciences and Applications

Some classical representation theoretical models exhibit similar asymptotic behavior with eigenvalues of random matrices. The goal of the current course is to present and explain such similarities. No previous knowledge of either subjects is required.

We focus on two basic examples: Plancherel measure, which arises from representation theory, and the Gaussian unitary ensemble, one of the most known random matrix models. We will verify that asymptotically the distribution of the rightmost eigenvalue (or rightmost particle) in both cases is governed by the famous Tracy-Widom distribution, and the corresponding random point process tends to the Airy process. We will discuss the general setting of determinantal point processes to which these models belong, and the convenient tools for studying these models, such as Fredholm and Toeplitz determinants and (discrete) Riemann-Hilbert problems, as well as a connection with the Painlevé equations.

The course is similar in name to the last year’s course by Anton Nazarov, but the topic is different: the latter focused on generalizations of Schur-Weyl dualities and the connection to random matrices via asymptotics of transition probabilities, while the current course focuses on a direct comparison of the models from both fields.