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 \(K\)

\[\iint K(x,y)^2 \mu(dx) \mu(dy) < \infty,\]
Self adjoint equiv. hermitian
\[K(x,y) = \overline{K(y,x)},\]Locally trace class
\[\forall B\subset \mathbb{X} \text{ compact}, \quad \int_B K(x,x) \mu(dx) < \infty.\]
Under assumptions 13, the
Hint
implies \(K\) to be a 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 such as the Carries process.
See also
Remarks 12 and Theorem 3 [Sos00]
Theorem 22 [HKPVirag06]
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
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
Lemma 21 [HKPVirag06]
Proposition 2.11 [Joh06] biorthogonal families