# Determinantal point processes

July 31, 2018 — March 16, 2021

linear algebra
Monte Carlo
point processes

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

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

The most popular tutorial introduction to this topic seems to be . I found it unhelpful as it is rooted in discrete-space problems which is precisely where I do not work. For continuous state space, and Terry Tao’s summary of are good.

Interesting property: The zeros random polynomials with Gaussian coefficients are apparently to be distributed as DPPs .

One idea these processes provoke is use as a source of random low-discrepancy samples for quadrature, which I have seen suggested by Richard Xu Qiao et al. (2016) and Belhadji, Bardenet, and Chainais (2019).

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