We study strong asymptotics of L^r extremal polynomials for measures supported on Jordan regions with C^1+ boundary for 0
We analyse several point processes on the sphere 𝕊^3⊂ℝ^4 by evaluating and comparing their discrete logarithmic energy: E_0(ω _N) = ∑ _[ k,j=1; k j ]^Nlog1/‖ x_k-x_j‖, ω _N={x_1,… ,x_N}⊂𝕊^3. Using the Hopf fibration, we lift several well-distributed processes from the 2-dimensional sphere–including i.i.d. uniform points, antipodally symmetric processes, and determinantal point processes –to 𝕊^3 and compare their expected energies. For the Spherical ensemble, a well-known determinantal point process on 𝕊^2 , we obtain a lifted process whose expected logarithmic energy provides the best known upper bound for the minimal logarithmic energy on 𝕊^3 .
In [Jalowy, Kabluchko, Marynych, Part I of the present work, Constr. Approx.], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to ∞ . In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little q-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.
Let A be an n × n generic and irreducible matrix. We consider power series expansions for eigenvectors corresponding to the real part of e^-iθA and use them to analyze and approximate the components of the Kippenhahn algebraic curve associated with the numerical range of A. We obtain information about the local behavior of the curve. As a result we obtain new equations for the series expansions, new conditions under which these components contain no line segments as well as conditions under which a component degenerates to a point. Applying these results to work of Mirman on compressions of unitary matrices, we show that in Mirman’s setting the components are locally asymptotically quadratic at all but finitely many points.
