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.
