Daniel Ofner: Exploring Orthogonal Polynomial Ensembles via Recurrence Relations

Abstract:

Determinantal Point Processes (DPPs) are stochastic point processes whose probability distributions can be expressed in terms of the determinant of a correlation kernel. In many cases, the correlation kernel of a given DPP can be represented as a squared Vandermonde determinant. By applying elementary column operations, this correlation kernel can be identified with the Christoffel–Darboux kernel, a well-known object in the theory of orthogonal polynomials. Such processes are commonly referred to as Orthogonal Polynomial Ensembles (OPEs). OPEs naturally arise in random matrix theory and statistical mechanics.

Two extensively studied special cases of these processes are the Circular Unitary Ensemble (CUE) and the Gaussian Unitary Ensemble (GUE). In this talk, we discuss the connection between DPPs and orthogonal polynomials, highlighting how known results about orthogonal polynomials can be leveraged to study the DPP corresponding to the OPE. In particular, we explore the well-known recurrence relations of the orthogonal polynomials and demonstrate that, through the bijection between the recurrence coefficients and the underlying measure, one can derive results regarding the asymptotic distribution of a given OPE.