A variational principle for correlation functions for unitary ensembles, with applications

In the theory of random matrices for unitary ensembles associated with Hermitian matrices, $m$ -point correlation functions play an important role. We show that they possess a useful variational principle. Let $μ$ be a measure with support in the real line, and $K_{n}$ be the $n$ -th reproducing kernel for the associated orthonormal polynomials. We prove that, for $m \geq 1$ ,

det {[K_{n} (μ, x_{i}, x_{j})]}_{1 \leq i, j \leq m} = m! sup_{P} \frac{P^{2} (\underline{x})}{\int P^{2} (\underline{t}) d μ^{\times m} (\underline{t})}

where the supremum is taken over all alternating polynomials $P$ of degree at most $n - 1$ in $m$ variables $\underline{x} = (x_{1}, x_{2}, \dots, x_{m})$ . Moreover, $μ^{\times m}$ is the $m$ -fold Cartesian product of $μ$ . As a consequence, the suitably normalized $m$ -point correlation functions are monotone decreasing in the underlying measure $μ$ . We deduce pointwise one-sided universality for arbitrary compactly supported measures, and other limits.

Keywords

orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions

Mathematical Subject Classification 2010

Primary: 15B52, 60B20, 60F99, 42C05, 33C50

Milestones

Received: 12 August 2011

Revised: 16 August 2011

Accepted: 13 February 2012

Published: 1 June 2013

Corrected: 30 July 2013 (equation (1-3))

Authors

Doron S. Lubinsky
School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332 United States

Vol. 6, No. 1, 2013

Doron S. Lubinsky

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors