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

    \[\iint_{\mathbb{X}\times \mathbb{X}} |K(x,y)|^2 \mu(dx) \mu(dy) < \infty,\]
  2. Self-adjoint equiv. Hermitian

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

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


    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


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


Under assumptions 1, 2, and 3

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

See also

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

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


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.

See also