Global fluctuations for Multiple Orthogonal Polynomial Ensembles

TL;DR

This paper establishes Central Limit Theorems for linear statistics in Multiple Orthogonal Polynomial Ensembles, including models like GUE with external source and Wishart matrices, using recurrence matrices and Toeplitz operator techniques.

Contribution

It introduces a novel approach using recurrence matrix limits and the Baker--Campbell--Hausdorff formula to prove CLTs for these ensembles, expanding understanding of their fluctuations.

Findings

Proves CLTs for linear statistics in multiple orthogonal polynomial ensembles.

Shows the recurrence matrix limits lead to Toeplitz operator representations.

Demonstrates the method on GUE with external source, Wishart matrices, and related measures.

Abstract

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others. Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker--Campbell--Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero. We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.