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

Lubinsky, Doron

doi:10.2140/apde.2013.6.109

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

Doron S. Lubinsky

Vol. 6 (2013), No. 1, 109–130

DOI: 10.2140/apde.2013.6.109

Abstract

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