Point process

A point process is a random locally finite configuration of points, usually represented as a random counting measure. A finite eigenvalue configuration of a random matrix is an example. Multiplicities and whether the process is simple should be specified.

Table of contents
- Determinantal point process Point process
- Correlation function of a point process Point process

Determinantal point process ( $R_{k} (x) = det [K (x_{i}, x_{j})]_{i, j = 1}^{k}$ )

A point process is determinantal with kernel

K

relative to a specified reference measure if every correlation function of a point process is the indicated determinant. A rank-

m

finite-rank projection kernel gives an exactly

m

-point process, whose symmetric joint density is

det [K (x_{i}, x_{j})] / m!

.

Correlation function of a point process ( $R_{k}$ )

The

k

th correlation function is the density of the

k

th factorial moment measure: integrating a nonnegative test function against

R_{k}

gives the expectation of its sum over ordered

k

-tuples of distinct points. For exactly

m

exchangeably labelled points with symmetric joint density

f_{m}

, it is

m! / (m - k)!

times the marginal density of

k

labels. It is not normalized to have total mass one.

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