Average Characteristic Polynomials of Determinantal Point Processes

TL;DR

This paper studies the average characteristic polynomial of determinantal point processes, establishing conditions for the convergence of zero distributions and applying these results to classical ensembles and free probability.

Contribution

It provides a sufficient condition for the zero distribution convergence of certain determinantal point processes and applies this to classical and multiple orthogonal polynomial ensembles.

Findings

Established conditions for zero distribution convergence.

Unified description of convergence for classical Orthogonal Polynomial Ensembles.

Derived limiting zero distributions for multiple Hermite and Laguerre polynomials.

Abstract

We investigate the average characteristic polynomial $\mathbb E\big[\prod_{i=1}^N(z-x_i)\big] $ where the $x_i$'s are real random variables which form a determinantal point process associated to a bounded projection operator. For a subclass of point processes, which contains Orthogonal Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as $N$ goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from a theorem of Kuijlaars and Van Assche a unified way to describe the almost sure convergence for classical Orthogonal Polynomial Ensembles. As another application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures.