Integrable probability: Random matrices at high and low temperatures
I will start by outlining what integrable probability is and then demonstrate its principles on a set of questions related to the random matrix theory.
The questions concern the dependence of eigenvalue distributions of random matrices on the parameter Beta, which takes values 1, 2, or 4, depending on whether we deal with real, complex, or quaternionic matrices. In the terminology of statistical mechanics, Beta is inverse-proportional to the temperature in the system and, as I will explain, this parameter can also take arbitrary positive real values. In the talk we will discuss a rich asymptotic theory as Beta tends to zero or to infinity.