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A variational principle for correlation functions for unitary ensembles, with applications. (English) Zbl 1281.15044

Let \(\mu\) be a positive measure on the real line with infinitely many points in its support, and \(\int x^j\, d\mu(x)\) finite for \(j=0,1,2,\dots\). Then we may define orthogonal polynomials \(p_n(x)\) (of degree \(n\) and positive coefficients) satisfying \(\int p_np_m\, d\mu=\delta_{mn}\). The \(n\)-th reproducing kernel is \(K_n(\mu,x_i,x_j)=\sum_{r=0}^{n-1}p_r(x_i)p_r(x_j)\).
The author proves in Theorem 1.1. that for \(m\geq 1\), \[ \det\left[K_n(\mu,x_i,x_j)\right]_{1\leq i,j\leq m}=m!\sup_P\frac{P^2(\underline{x})}{\int P^2(\underline{t})d\mu^{\times m}(\underline{t})} \] where the supremum is taken over all alternating polynomials \(P\) of degree of at most \(n-1\) in \(m\) variables \(\underline{x}=(x_1,x_2,\dots,x_m)\).
Corollary 1.2 states that \(R_m^n(\mu;x_1,x_2,\dots,x_m)=\det\left[K_n(\mu,x_i,x_j)\right]_{1\leq i,j\leq m}\) is a monotone decreasing function of \(\mu\) and a monotone increasing function of \(n\).
Applications of Corollary 1.2 to asymptotics and universality limits are given in Section 2.

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
62H20 Measures of association (correlation, canonical correlation, etc.)
15A15 Determinants, permanents, traces, other special matrix functions
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