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Given a set , a (simple) point process is a random subset of . (A non-simple point process would allow multiplicity; more formally, is no longer a subset of , but is a Radon measure on , where we give the structure of a locally compact Polish space, but I do not wish to dwell on these sorts of technical issues here.) Typically, will be finite or countable, even when is uncountable. Basic examples of point processes include
- (Bernoulli point process) is an at most countable set, is a parameter, and a random set such that the events for each are jointly independent and occur with a probability of each. This process is automatically simple.
- (Discrete Poisson point process) is an at most countable space, is a measure on (i.e. an assignment of a non-negative number to each ), and is a multiset where the multiplicity of in is a Poisson random variable with intensity , and the multiplicities of as varies in are jointly independent. This process is usually not simple.
- (Continuous Poisson point process) is a locally compact Polish space with a Radon measure , and for each of finite measure, the number of points that contains inside is a Poisson random variable with intensity . Furthermore, if are disjoint sets, then the random variables are jointly independent. (The fact that Poisson processes exist at all requires a non-trivial amount of measure theory, and will not be discussed here.) This process is almost surely simple iff all points in have measure zero.
- (Spectral point processes) The spectrum of a random matrix is a point process in (or in , if the random matrix is Hermitian). If the spectrum is almost surely simple, then the point process is almost surely simple. In a similar spirit, the zeroes of a random polynomial are also a point process.
A remarkable fact is that many natural (simple) point processes are determinantal processes. Very roughly speaking, this means that there exists a positive semi-definite kernel such that, for any , the probability that all lie in the random set is proportional to the determinant . Examples of processes known to be determinantal include non-intersecting random walks, spectra of random matrix ensembles such as GUE, and zeroes of polynomials with gaussian coefficients.
I would be interested in finding a good explanation (even at the heuristic level) as to why determinantal processes are so prevalent in practice. I do have a very weak explanation, namely that determinantal processes obey a large number of rather pretty algebraic identities, and so it is plausible that any other process which has a very algebraic structure (in particular, any process involving gaussians, characteristic polynomials, etc.) would be connected in some way with determinantal processes. I’m not particularly satisfied with this explanation, but I thought I would at least describe some of these identities below to support this case. (This is partly for my own benefit, as I am trying to learn about these processes, particularly in connection with the spectral distribution of random matrices.) The material here is partly based on this survey of Hough, Krishnapur, Peres, and Virág.