# 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, 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 K2 → 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

$\rho _{n}(x_{1},\ldots ,x_{n})=\det[K(x_{i},x_{j})]_{1\leq i,j\leq n}$ for every n ≥ 1 and x1, . . . , xn ∈ Λ.

## Properties

### Existence

The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities ρk.

$\rho _{k}(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\rho _{k}(x_{1},\ldots ,x_{k})\quad \forall \sigma \in S_{k},k$ • Positivity: For any N, and any collection of measurable, bounded functions φk:Λk → ℝ, k = 1,. . . ,N with compact support:
If
$\quad \varphi _{0}+\sum _{k=1}^{N}\sum _{i_{1}\neq \cdots \neq i_{k}}\varphi _{k}(x_{i_{1}}\ldots x_{i_{k}})\geq 0{\text{ for all }}k,(x_{i})_{i=1}^{k}$ Then
$\quad \varphi _{0}+\sum _{k=1}^{N}\int _{\Lambda ^{k}}\varphi _{k}(x_{1},\ldots ,x_{k})\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d}}x_{1}\cdots {\textrm {d}}x_{k}\geq 0{\text{ for all }}k,(x_{i})_{i=1}^{k}$ ### 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.

## Examples

### 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 partitions of integers (and therefore on Young diagrams) 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 ℤ[clarification needed] + ​12 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 → 2(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 2(E) spanned by star flows. Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

$K(e,f)=\langle I^{e},I^{f}\rangle ,\quad e,f\in E$ .