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

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 × n matrix. The eigenvalues pⱼ of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function |G(x,n)|:=#:pⱼ>Ce-x n\, ($C>0$ here is a fixed constant) and establish the asymptotic behavior of its average over the interval x ∈ (λ-ε, λ+ε) by relating the function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on the unit circle S¹, which is of independent interest. Namely, for given θ, 0<θ< 2 π, and given q, $0<q<1$, we determine the function J(y) =inf (μ): μ ∈ P(S¹), μ(Aθ) = y\, where I(μ):= log1/|z - ζ| dμ(z) dμ(ζ) is the logarithmic energy of a probability measure μ supported on the unit circle and Aθ is the arc from e-i θ/2 to ei θ/2.