I will start with the definition of quasi-infinitely distributions and will exhibit some of their properties. Afterwards, I'll give and analyze two discrete examples constructed from the eigenspaces of the magnetic laplacians in the complex plane and in the hyperbolic disc. Doing so leads to new Lévy processes. Finally, I will show how one can use the reproducing kernels of the aforementioned eigenspaces to define new determinantal point processes and study their hyperuniformity.