Analysis & PDE

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

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

Doron Lubinsky

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15B52: Random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F99: None of the above, but in this section 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable

orthogonal polynomials random matrices unitary ensembles correlation functions Christoffel functions


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.

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