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(q) 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 \(I(\mu ):= \int \!\int \log \frac{1}{|z - \zeta |} d\mu (z) d\mu (\zeta )\) 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 \(e^{i \theta /2}\).
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L. Kryvonos and E.B. Saff wrote the main manuscript, all authors reviewed the manuscript.
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The problem was communicated to the second author shortly before the untimely death of Professor Meckes.
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Kryvonos, L., Saff, E.B. On a problem of E. Meckes for the unitary eigenvalue process on an arc. Anal.Math.Phys. 14, 59 (2024). https://doi.org/10.1007/s13324-024-00919-w
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DOI: https://doi.org/10.1007/s13324-024-00919-w
Keywords
- Unitary eigenvalue process
- Discrete prolate spheroidal wave functions
- Logarithmic energy problem with constraints