Comment by CrazyStat

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