# Determinantal point processes

Placeholder notes for a version of the point process, with which I am completely unfamiliar, but about which I am incidentally curious.

Tao is wonderful as ever

Wikipedia says

Let $$\Lambda$$ be a locally compact Polish space and $$\mu$$ be a Radon measure on $$\Lambda$$.
Also, consider a measurable function $$K:\Lambda^2\rightarrow \mathbb{C}$$.

We say that $$X$$ is a *determinantal point process* on
$$\Lambda$$ with kernel $$K$$ if it is a simple point process on $$\Lambda$$
with a joint intensity/Factorial_moment_densityorcorrelation function
(which is the density of its factorial moment measure) given by

$\rho_n(x_1,\ldots,x_n) = \det[K(x_i,x_j)]_{1 \le i,j \le n}$

for every $$n\ geq 1$$ and $$x_1,\dots, x_n\in \Lambda.$$