Vol. 6, No. 1, 2013

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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

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[Kn(μ,xi,xj)]1i,jm = m!supP P2(x ¯) P2(t ¯)dμ×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.

orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Mathematical Subject Classification 2010
Primary: 15B52, 60B20, 60F99, 42C05, 33C50
Received: 12 August 2011
Revised: 16 August 2011
Accepted: 13 February 2012
Published: 1 June 2013
Corrected: 30 July 2013 (equation (1-3))
Doron S. Lubinsky
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332
United States