The Diamond ensemble: a very nice constructive set of points on the 2-sphere.
Ujue Etayo (Universidad de Cantabria)
given at NuHAG seminar (17.10.18 11:30)
Sets of points on the 2-sphere that are well-distributed in some sense conform an compelling object of study, see for example [Brauchart and Grabner, 2015] for an interesting survey with different approaches to well-distributed points. Constructive families of points in the 2-sphere share a common problem: it is extremely hard to compute the asymptotics when the number of points goes to infinity for some quantities such as the discrepancy or the Riesz energies. In [Hardin et al., 2016] you can find a nice survey on constructive sets of points.
In this talk we present the Diamond ensemble [BeltrÃ¡n and Etayo, 2018], a family of random sets of points on the 2-sphere depending on several parameters. For different values of the parameters we obtain some sets of points that are extremely similar to other well known families of points such as the octhaedral points (see [Holhos and Rosca, 2014]) or the zonal equal area nodes (see [Rakhmanov et al., 1994]).
The most important property of the Diamond ensemble is that, for some of these parameters, the asymptotic expected value of the logarithmic energy of the points can be computed rigorously and shown to attain very small values, quite close to the conjectured minimal value, see [Brauchart et al., 2012] and [BÃ©termin and Sandier, 2018].