Energy and random point processes on two-point homogeneous manifolds

Programa de Doctorat en Matemàtiques i Informàtica [eng] We study discrete energy minimization problems on two-point homogeneous manifolds. Since finding N-point configurations with optimal energy is highly challenging, recent approaches have involved examining random point processes with low expected energy to obtain good N- point configurations. In Chapter 2, we compute the second joint intensity of the random point process given by the zeros of elliptic polynomials, which enables us to recover the expected logarithmic energy on the 2-dimensional sphere previously computed by Armentano, Beltrán, and Shub. Moreover, we obtain the expected Riesz s-energy, which is remarkably close to the conjectured optimal energy. The expected energy serves as a bound for the extremal s-energy, thereby improving upon the bounds derived from the study of the spherical ensemble by Alishahi and Zamani. Among other additional results, we get a closed expression for the expected separation distance between points sampled from the zeros of elliptic polynomials. In Chapter 3, we explore the average discrepancies and worst-case errors of random point configurations on the d-dimensional sphere. We find that the points drawn from the so called spherical ensemble and the zeros of elliptic polynomials achieve optimal spherical L^2 cap discrepancy on average. Additionally, we provide an upper bound for the L^intiy discrepancy for N-point configurations drawn from the harmonic ensemble on any two-point homogeneous space, thereby generalizing the previous findings for the sphere by Beltrán, Marzo and Ortega- Cerdà. We introduce a nondeterministic version of the Quasi Monte Carlo (QMC) strength for random sequences of points and compute its value for the spherical ensemble, the zeros of elliptic polynomials, and the harmonic ensemble. Finally, we compare our results with the conjectured QMC strengths of certain deterministic distributions associated with these random point processes. In Chapter 4, our focus hits to the Green energy minimization problem. Firstly, we extend the work by Beltrán and Lizarte on spheres to establish a close to sharp lower bound for the minimal Green energy on any two-point homogeneous manifold, improving on the existing lower bounds on projective spaces. Secondly, by adapting a method introduced by Wolff, we deduce an upper bound for the L^intiy discrepancy of N-point sets that minimize the Green energy. [cat] Estudiem problemes de minimització d'energia discreta en v