A Variational Principle for Correlation Functions for Unitary Ensembles, with Applications

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 nth reproducing kernel for the associated orthonormal polynomials. We prove that for m ≥ 1, det [Kn (μ, xi, xj)]1≤1,j≤m = m! sup P P 2 (x) ∫ P 2 (t) dμ×m (t) where the sup is taken over all alternating polynomials P of degree ≤ 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, RandomMatrices, Unitary Ensembles, Correlation Functions, Christoffel functions. 15B52, 60B20, 60F99, 42C05, 33C50