Determinantal point process

In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, physics, machine learning, and wireless network modeling.

Definition
Let $$\Lambda$$ be a locally compact Polish space and $$\mu$$ be a Radon measure on $$\Lambda$$. Also, consider a measurable function $$K: \Lambda^2 \to \mathbb{C}$$.

We say that $$X$$ is a determinantal point process on $$\Lambda$$ with kernel $$K$$ if it is a simple point process on $$\Lambda$$ with a joint intensity or correlation function (which is the density of its factorial moment measure) given by


 * $$ \rho_n(x_1,\ldots,x_n) = \det[K(x_i,x_j)]_{1 \le i,j \le n} $$

for every n ≥ 1 and x1, ..., xn ∈ Λ.

Existence
The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities ρk.
 * Symmetry: ρk is invariant under action of the symmetric group Sk. Thus: $$\rho_k(x_{\sigma(1)}, \ldots, x_{\sigma(k)}) = \rho_k(x_1, \ldots, x_k)\quad \forall \sigma \in S_k, k$$
 * Positivity: For any N, and any collection of measurable, bounded functions $\varphi_k : \Lambda^k \to \mathbb{R}$, k = 1, ..., N with compact support: If $$ \varphi_0 + \sum_{k=1}^N \sum_{i_1 \neq \cdots \neq i_k } \varphi_k(x_{i_1} \ldots x_{i_k})\ge 0 \text{ for all }k,(x_i)_{i = 1}^k $$ Then $$ \varphi_0 + \sum_{k=1}^N \int_{\Lambda^k} \varphi_k(x_1, \ldots, x_k)\rho_k(x_1,\ldots,x_k)\,\textrm{d}x_1\cdots\textrm{d}x_k \ge0 \text{ for all } k, (x_i)_{i = 1}^k $$

Uniqueness
A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρk is $$\sum_{k = 0}^\infty \left( \frac{1}{k!} \int_{A^k} \rho_k(x_1,\ldots,x_k) \, \textrm{d}x_1\cdots\textrm{d}x_k \right)^{-\frac{1}{k}} = \infty$$ for every bounded Borel A ⊆ Λ.

Gaussian unitary ensemble
The eigenvalues of a random m × m Hermitian matrix drawn from the Gaussian unitary ensemble (GUE) form a determinantal point process on $$\mathbb{R}$$ with kernel
 * $$K_m(x,y) = \sum_{k=0}^{m-1} \psi_k(x) \psi_k(y)$$

where $$\psi_k(x)$$ is the $$k$$th oscillator wave function defined by

$$ \psi_k(x)= \frac{1}{\sqrt{\sqrt{2n}n!}}H_k(x) e^{-x^2/4} $$

and $$H_k(x)$$ is the $$k$$th Hermite polynomial.

Poissonized Plancherel measure
The poissonized Plancherel measure on integer partition (and therefore on Young diagramss) plays an important role in the study of the longest increasing subsequence of a random permutation. The point process corresponding to a random Young diagram, expressed in modified Frobenius coordinates, is a determinantal point process on $$\mathbb{Z}$$ + $1/undefined$ with the discrete Bessel kernel, given by:

$$K(x,y) = \begin{cases} \sqrt{\theta} \, \dfrac{k_+(|x|,|y|)}{|x|-|y|} & \text{if } xy >0,\\[12pt] \sqrt{\theta} \, \dfrac{k_-(|x|,|y|)}{x-y} & \text{if } xy <0, \end{cases} $$ where $$ k_+(x,y) = J_{x-\frac{1}{2}}(2\sqrt{\theta})J_{y+\frac{1}{2}}(2\sqrt{\theta}) - J_{x+\frac{1}{2}}(2\sqrt{\theta})J_{y-\frac{1}{2}}(2\sqrt{\theta}), $$ $$ k_-(x,y) = J_{x-\frac{1}{2}}(2\sqrt{\theta})J_{y-\frac{1}{2}}(2\sqrt{\theta}) + J_{x+\frac{1}{2}}(2\sqrt{\theta})J_{y+\frac{1}{2}}(2\sqrt{\theta}) $$ For J the Bessel function of the first kind, and θ the mean used in poissonization.

This serves as an example of a well-defined determinantal point process with non-Hermitian kernel (although its restriction to the positive and negative semi-axis is Hermitian).

Uniform spanning trees
Let G be a finite, undirected, connected graph, with edge set E. Define Ie:E → ℓ2(E) as follows: first choose some arbitrary set of orientations for the edges E, and for each resulting, oriented edge e, define Ie to be the projection of a unit flow along e onto the subspace of ℓ2(E) spanned by star flows. Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel
 * $$K(e,f) = \langle I^e,I^f \rangle ,\quad e,f \in E$$.