On a problem of E. Meckes for the unitary eigenvalue process on an arc

Abstract: We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random $n \times n$ matrix. The eigenvalues $p_{j}$ of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function $|G(x,n)|:=#{j:p_j>Ce^{-x n}}$, ($C>0$ here is a fixed constant) and establish the asymptotic behavior of its average over the interval $x \in (\lambda-\varepsilon, \lambda+\varepsilon)$ by relating the function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on the unit circle $S^{1}$, which is of independent interest. Namely, for given $\theta$, $0<\theta< 2 \pi$, and given $q$, $0<q<1$, we determine the function $J(q) =\inf {I(\mu): \mu \in \mathcal{P}(S^{1}), \mu(A_{\theta}) = q}$, where $I(\mu):= \iint \log\frac{1}{|z - \zeta|} d\mu(z) d\mu(\zeta)$ is the logarithmic energy of a probability measure $\mu$ supported on the unit circle and $A_{\theta}$ is the arc from $e^{-i \theta/2}$ to $e^{i \theta/2}$.