I will compare some two-dimensional point processes using three criteria of “order”: 1) growth of the number variance in large disks (hyperuniformity) 2) "rigidity" à la Ghosh-Peres 3) transportation cost to the Lebesgue measure. Existing examples range from "very ordered" (a lattice) to "not ordered at all" (a cloud of independent points), and I will focus on the zeroes of the Gaussian Analytic Function (as studied by Nazarov-Sodin-Volberg) and the 2d Coulomb gas, which exhibit somewhat surprising properties.
I will also mention a few known links between those three notions, as well as open questions.
