Definition¶
Point Process¶
Let \(\mathbb{X} = \mathbb{R}^d, \mathbb{C}^d \text{ or } \mathbb{S}^{d1}\) 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
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
See also
Existence¶
One can view \(K\) as an integral operator on \(L^2(\mu)\)
To access spectral properties of the kernel, it is common practice to assume \(K\)

\[\iint_{\mathbb{X}\times \mathbb{X}} K(x,y)^2 \mu(dx) \mu(dy) < \infty,\]
Selfadjoint equiv. Hermitian
\[K(x,y) = \overline{K(y,x)},\]Locally trace class
\[\forall B\subset \mathbb{X} \text{ compact}, \quad \int_B K(x,x) \mu(dx) < \infty.\]
Hint
implies \(K\) to be a continuous compact operator.
with 1. allows to apply the spectral theorem, providing
\[K(x,y) = \sum_{n=0}^{\infty} \lambda_n \phi_{n}(x)\phi_{n}(y), \quad \text{where } K\phi_{n} = \lambda_n \phi_{n}.\]
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
See also
Remarks 12 and Theorem 3 [Sos00]
Theorem 22 [HKPVirag06]
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
Construction¶
A canonical way to construct DPPs generating configurations of at most \(N\) points is the following.
Consider \(N\) orthonormal functions \(\phi_{0},...,\phi_{N−1} \in L^2(\mu)\)
and attach \([0,1]\)valued coefficients \(\lambda_n\) such that
The special case where \(\lambda_0=\cdots=\lambda_{N1}=1\) corresponds to the construction of a projection DPP with \(N\) points.
See also
Lemma 21 [HKPVirag06]
Proposition 2.11 [Joh06] biorthogonal families