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2013 A variational principle for correlation functions for unitary ensembles, with applications
Doron Lubinsky
Anal. PDE 6(1): 109-130 (2013). DOI: 10.2140/apde.2013.6.109


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 m1,

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 n1 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.


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Doron Lubinsky. "A variational principle for correlation functions for unitary ensembles, with applications." Anal. PDE 6 (1) 109 - 130, 2013.


Received: 12 August 2011; Revised: 16 August 2011; Accepted: 13 February 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1281.15044
MathSciNet: MR3068541
Digital Object Identifier: 10.2140/apde.2013.6.109

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

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

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.6 • No. 1 • 2013
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