# Definition¶

## Point Process¶

Let $$\mathbb{X} = \mathbb{R}^d, \mathbb{C}^d \text{ or } \mathbb{S}^{d-1}$$ be the ambient space, we endow it with the corresponding Borel $$\sigma$$-algebra $$\mathcal{B}(\mathbb{X})$$ together with a reference measure $$\mu$$.

For our purpose, we consider point processes as locally finite random subsets $$\mathcal{X} \subset \mathbb{X}$$ i.e.

$\forall C \subset \mathbb{X} \text{ compact}, \quad \#(\mathcal{X} \cap C) < \infty.$

Hint

A point process is a random subset of points $$\mathcal{X} \triangleq\{X_1, \dots, X_N\} \subset \mathbb{X}$$ with $$N$$ being random.

More formal definitions can be found in [MollerW04] Section 2 and [Joh06] Section 2 and bibliography therein.

To understand the interaction between the points of a point process, one focuses on the interaction of each cloud of $$k$$ points (for all $$k$$). The corresponding $$k$$-correlation functions characterize the underlying point process.

## Correlation functions¶

For $$k\geq 0$$, the $$k$$-correlation function $$\rho_k$$ is defined by:

$$\forall f : \mathbb{X}^k \to \mathbb{C}$$ bounded measurable

$\begin{split}\mathbb{E} \left[ \sum_{ \substack{ (X_1,\dots,X_k) \\ X_1 \neq \dots \neq X_k \in \mathcal{X}} } f(X_1,\dots,X_k) \right] = \int_{\mathbb{X}^k} f(x_1,\dots,x_k) \rho_k(x_1,\dots,x_k) \prod_{i=1}^k \mu(dx_i).\end{split}$

Hint

The $$k$$-correlation function does not always exists, but but when they do, they have a meaningful interpretation.

$\begin{split}" \rho_k(x_1,\dots,x_k) \mu(dx_{1}), \dots, \mu(dx_{N}) = \mathbb{P} \left[ \begin{array}{c} \text{there is 1 point in each}\\ B(x_1, d x_1), \dots, B(x_n, d x_n) \end{array} \right] ",\end{split}$

where $$B(x, dx)$$ denotes the ball centered at $$x$$ with radius $$dx$$.

A Determinant Point Process (DPP) is a point process on $$(\mathbb{X}, \mathcal{B}(\mathbb{X}), \mu)$$ parametrized by a kernel $$K$$ associated to the reference measure $$\mu$$. The $$k$$-correlation functions read

$\forall k\geq 1, \quad \rho_k(x_1,\dots,x_k) = \det [K(x_i, x_j)]_{i,j=1}^k.$

## Existence¶

One can view $$K$$ as an integral operator on $$L^2(\mu)$$

$\forall x \in \mathbb{X}, Kf(x) = \int_{\mathbb{X}} K(x,y) f(y) \mu(dy).$

To access spectral properties of the kernel, it is common practice to assume $$K$$

1. Hilbert Schmidt

$\iint_{\mathbb{X}\times \mathbb{X}} |K(x,y)|^2 \mu(dx) \mu(dy) < \infty,$

$K(x,y) = \overline{K(y,x)},$
3. Locally trace class

$\forall B\subset \mathbb{X} \text{ compact}, \quad \int_B K(x,x) \mu(dx) < \infty.$

Hint

1. implies $$K$$ to be a continuous compact operator.

1. with 1. allows to apply the spectral theorem, providing

$K(x,y) = \sum_{n=0}^{\infty} \lambda_n \phi_{n}(x)\phi_{n}(y), \quad \text{where } K\phi_{n} = \lambda_n \phi_{n}.$
1. makes sure there is no accumulation of points: $$|\mathcal{X}\cap B| = \int_B K(x,x) \mu(dx) \leq \infty$$, see also Number of points

Warning

These are only sufficient conditions, there indeed exist DPPs with non symmetric kernels, see, e.g., Carries process.

Important

Under assumptions 1, 2, and 3

$\operatorname{DPP}(K) \text{ exists} \Longleftrightarrow 0\leq \lambda_n \leq 1, \quad \forall n \in \mathbb{N}$

## Projection DPPs¶

$$\operatorname{DPP}(K)$$ is said to be a projection DPP with reference measure $$\mu$$ when $$K:\mathbb{X}\times \mathbb{X}\to \mathbb{C}$$ is a orthogonal projection kernel, that is

$K(x,y)=\overline{K(y,x)} \quad\text{and}\quad \int_{\mathbb{X}} K(x, z) K(z, y) \mu(d z) = K(x, y)$

## Construction¶

A canonical way to construct DPPs generating configurations of at most $$N$$ points is the following.

Consider $$N$$ orthonormal functions $$\phi_{0},...,\phi_{N−1} \in L^2(\mu)$$

$\int \phi_{k}(x)\phi_{l}(x)\mu(dx) = \delta_{kl},$

and attach $$[0,1]$$-valued coefficients $$\lambda_n$$ such that

$K(x, y) = \sum_{n=0}^{N-1} \lambda_n \phi_{n}(x)\phi_{n}(y).$

The special case where $$\lambda_0=\cdots=\lambda_{N-1}=1$$ corresponds to the construction of a projection DPP with $$N$$ points.