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