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
Abstract

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  $\mu$ be a measure with support in the real line, and ${K}_{n}$ be the $n$-th reproducing kernel for the associated orthonormal polynomials. We prove that, for $m\ge 1$,

 $det{\left[{K}_{n}\left(\mu ,{x}_{i},{x}_{j}\right)\right]}_{1\le i,j\le m}=m!\underset{P}{sup}\frac{{P}^{2}\left(\underset{¯}{x}\right)}{\int {P}^{2}\left(\underset{¯}{t}\right)\phantom{\rule{0.3em}{0ex}}d{\mu }^{×m}\left(\underset{¯}{t}\right)}$

where the supremum is taken over all alternating polynomials $P$ of degree at most $n-1$ in $m$ variables $\underset{¯}{x}=\left({x}_{1},{x}_{2},\dots ,{x}_{m}\right)$. Moreover, ${\mu }^{×m}$ is the $m$-fold Cartesian product of $\mu$. As a consequence, the suitably normalized $m$-point correlation functions are monotone decreasing in the underlying measure  $\mu$. 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
Mathematical Subject Classification 2010
Primary: 15B52, 60B20, 60F99, 42C05, 33C50