## Analysis & PDE

• Anal. PDE
• Volume 6, Number 1 (2013), 109-130.

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

Doron Lubinsky

#### 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 $Kn$ be the $n$-th reproducing kernel for the associated orthonormal polynomials. We prove that, for $m≥1$,

$det [ K n ( μ , x i , x j ) ] 1 ≤ i , j ≤ m = m ! sup P P 2 ( x ¯ ) ∫ P 2 ( t ¯ ) d μ × m ( t ¯ )$

where the supremum is taken over all alternating polynomials $P$ of degree at most $n−1$ in $m$ variables $x¯=(x1,x2,…,xm)$. Moreover, $μ×m$ 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.

#### Article information

Source
Anal. PDE, Volume 6, Number 1 (2013), 109-130.

Dates
Revised: 16 August 2011
Accepted: 13 February 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731291

Digital Object Identifier
doi:10.2140/apde.2013.6.109

Mathematical Reviews number (MathSciNet)
MR3068541

Zentralblatt MATH identifier
1281.15044

#### Citation

Lubinsky, Doron. A variational principle for correlation functions for unitary ensembles, with applications. Anal. PDE 6 (2013), no. 1, 109--130. doi:10.2140/apde.2013.6.109. https://projecteuclid.org/euclid.apde/1513731291

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