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.

See also

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

    \[\iint |K(x,y)|^2 \mu(dx) \mu(dy) < \infty,\]
  2. Self adjoint equiv. hermitian

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

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

Under assumptions 1-3, the

\[\operatorname{DPP}(K) \text{ exists} \Longleftrightarrow \lambda_n \in [0,1], \quad \forall n.\]

Hint

    1. implies \(K\) to be a 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 such as the Carries process.

See also

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

\[K_N (x, y) = \sum_{n=0}^{N-1} \lambda_n \phi_{n}(x)\phi_{n}(y).\]

Note

In this setting, in order to generate configurations \(\{x_1, \dots ,x_N\}\) of \(N\) points a.s. set \(\lambda_n=1\). The corresponding kernel \(K_N\) is the projection onto \(\operatorname{Span} \{\phi_{0},...,\phi_{N−1}\}\)

See also