Comment by wduquette
1 month ago
The article has a graphic contrasting a "Random" distribution vs. a "Universal" distribution vs. a "Periodic" distribution. I'm guessing the "Random" distribution is actually a Poisson distribution, as that arises naturally in several cases.
But the big question is, does this "Universal" distribution match up to any well known probability distribution? Or could it be described by a relatively simple probability distribution function?
I think you mean a Poisson process rather than a Poisson distribution. The Poisson distribution is a discrete distribution on the non-negative integers. The Poisson process’s defining characteristic is that the number of points in any interval follows the Poisson distribution.
There have been a large variety of point processes explored in the literature, including some with repulsion properties that give this type of “universality” property. Perhaps unsurprisingly one way to do this is create your point process by taking the eigenvalues of a random matrix, which falls within the class of determinantal point processes [1]. Gibbs point processes are another important class.
[1] https://en.wikipedia.org/wiki/Determinantal_point_process
Just a layman: the graphic suggested to me that you might take the lines and their deviation from a periodic distribution. The random distribution is clearly further from periodic, the universal one closer. I wondered if there was some threshold that determined random vs. universal.