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.
"A variational principle for correlation functions for unitary ensembles, with applications." Anal. PDE 6 (1) 109 - 130, 2013. https://doi.org/10.2140/apde.2013.6.109