Integrable Systems and Random Matrix Theory Seminar

In this talk, we will consider the eigenvalue point processes of one-dimensional random Schrodinger operators(RSOs). RSOs arise naturally in many problems of mathematical physics, a particularly interesting example to random matrix community is the stochastic Airy operator whose eigenvalues point process(known as Airy beta process) appear as the soft edge scaling limits of beta ensembles.

We will mainly focus on a certain property of the eigenvalue point processes called number rigidity, introduced by Ghosh and Peres, which roughly states that the total number of points inside any compact set is deterministic conditioning on the configuration outside. A large class of random point processes are shown to be number rigid including most point processes arise as the scaling limits of eigenvalues of random matrices with a determinantal/Pfaffian structure. Here we give the first few examples of number rigidity of point processes whose two-point correlations are not known(even asymptotically). The main techniques is the semigroup approach developed by Gorin- Shkolnikov, Gaudreau Lamarre and this work, which express the Laplace transform of eigenvalue point processes of RSOs as functionals of Brownian motion through a Feymann-Kac formula. This is joint work with Pierre Yves Gaudreau Lamarre and Promit Ghosal. Speaker(s): Yuchen Liao (University of Michigan)