A variational principle for correlation functions for unitary ensembles, with applications (Q371157)

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 \(K_n(\mu,x_i,x_j)=\sum_{r=0}^{n-1}p_r(x_i)p_r(x_j)\).NEWLINENEWLINEThe author proves in Theorem 1.1. that for \(m\geq 1\), NEWLINE\[NEWLINE\det\left[K_n(\mu,x_i,x_j)\right]_{1\leq i,j\leq m}=m!\sup_P\frac{P^2(\underline{x})}{\int P^2(\underline{t})d\mu^{\times m}(\underline{t})}NEWLINE\]NEWLINE where the supremum is taken over all alternating polynomials \(P\) of degree of at most \(n-1\) in \(m\) variables \(\underline{x}=(x_1,x_2,\dots,x_m)\).NEWLINENEWLINECorollary 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 \(n\).NEWLINENEWLINEApplications of Corollary 1.2 to asymptotics and universality limits are given in Section 2.