The Living Thing / Notebooks :

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 â„‚.

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 $x_1,\dots, x_n\in \Lambda.$