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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 \(x_1,\dots, x_n\in \Lambda.\)