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.

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


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

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


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

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


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

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


    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


These are only sufficient conditions, there indeed exist DPPs with non symmetric kernels such as the Carries process.

See also


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

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


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