Details

Autor(en) / Beteiligte

• Kryvonos, L.
• Saff, E. B.

Titel

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

Ist Teil von

• Analysis and mathematical physics, 2024-06, Vol.14 (3), Article 59

Ort / Verlag

Cham: Springer International Publishing

Erscheinungsjahr

2024

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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 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 > C e - 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 ( q ) of the following energy problem on the unit circle S 1 , which is of independent interest. Namely, for given θ , 0 < θ < 2 π , and given q , 0 < q < 1 , we determine the function J ( q ) = inf { I ( μ ) : μ ∈ P ( S 1 ) , μ ( A θ ) = q } , where I ( μ ) : = ∫ ∫ log 1 | z - ζ | d μ ( z ) d μ ( ζ ) is the logarithmic energy of a probability measure μ supported on the unit circle and A θ is the arc from e - i θ / 2 to e i θ / 2 .