Analysis & PDE
- Anal. PDE
- Volume 6, Number 1 (2013), 109-130.
A variational principle for correlation functions for unitary ensembles, with applications
In the theory of random matrices for unitary ensembles associated with Hermitian matrices, -point correlation functions play an important role. We show that they possess a useful variational principle. Let be a measure with support in the real line, and be the -th reproducing kernel for the associated orthonormal polynomials. We prove that, for ,
where the supremum is taken over all alternating polynomials of degree at most in variables . Moreover, is the -fold Cartesian product of . As a consequence, the suitably normalized -point correlation functions are monotone decreasing in the underlying measure . We deduce pointwise one-sided universality for arbitrary compactly supported measures, and other limits.
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
Permanent link to this document
Digital Object Identifier
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
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