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,^[1] and wireless network modeling.^[2][3][4]

Definition

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

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

${\displaystyle \rho _{n}(x_{1},\ldots ,x_{n})=\det[K(x_{i},x_{j})]_{1\leq i,j\leq n))$

for every n ≥ 1 and x₁, ..., x_n ∈ Λ.^[5]

Properties

Existence

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

• Symmetry: ρ_k is invariant under action of the symmetric group S_k. Thus:
  $${\displaystyle \rho _{k}(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\rho _{k}(x_{1},\ldots ,x_{k})\quad \forall \sigma \in S_{k},k}$$

  
• Positivity: For any N, and any collection of measurable, bounded functions ${\displaystyle \varphi _{k}:\Lambda ^{k}\to \mathbb {R} }$, k = 1, ..., N with compact support:
  If
  $${\displaystyle \varphi _{0}+\sum _{k=1}^{N}\sum _{i_{1}\neq \cdots \neq i_{k))\varphi _{k}(x_{i_{1))\ldots x_{i_{k)))\geq 0{\text{ for all ))k,(x_{i})_{i=1}^{k))$$

  
  Then ^[6]
  $${\displaystyle \varphi _{0}+\sum _{k=1}^{N}\int _{\Lambda ^{k))\varphi _{k}(x_{1},\ldots ,x_{k})\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d))x_{1}\cdots {\textrm {d))x_{k}\geq 0{\text{ for all ))k,(x_{i})_{i=1}^{k))$$

  

Uniqueness

A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρ_k is

$${\displaystyle \sum _{k=0}^{\infty }\left({\frac {1}{k!))\int _{A^{k))\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d))x_{1}\cdots {\textrm {d))x_{k}\right)^{-{\frac {1}{k))}=\infty }$$

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 ${\displaystyle \mathbb {R} }$ with kernel

${\displaystyle K_{m}(x,y)=\sum _{k=0}^{m-1}\psi _{k}(x)\psi _{k}(y)}$

where ${\displaystyle \psi _{k}(x)}$ is the ${\displaystyle k}$th oscillator wave function defined by

$${\displaystyle \psi _{k}(x)={\frac {1}{\sqrt ((\sqrt {2n))n!))}H_{k}(x)e^{-x^{2}/4))$$

and ${\displaystyle H_{k}(x)}$ is the ${\displaystyle k}$th Hermite polynomial. ^[7]

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 ${\displaystyle \mathbb {Z} }$ + 1⁄2 with the discrete Bessel kernel, given by:

$${\displaystyle K(x,y)={\begin{cases}{\sqrt {\theta ))\,{\dfrac {k_{+}(|x|,|y|)}{|x|-|y|))&{\text{if ))xy>0,\\[12pt]{\sqrt {\theta ))\,{\dfrac {k_{-}(|x|,|y|)}{x-y))&{\text{if ))xy<0,\end{cases))}$$

$${\displaystyle k_{+}(x,y)=J_{x-{\frac {1}{2))}(2{\sqrt {\theta )))J_{y+{\frac {1}{2))}(2{\sqrt {\theta )))-J_{x+{\frac {1}{2))}(2{\sqrt {\theta )))J_{y-{\frac {1}{2))}(2{\sqrt {\theta ))),}$$

$${\displaystyle k_{-}(x,y)=J_{x-{\frac {1}{2))}(2{\sqrt {\theta )))J_{y-{\frac {1}{2))}(2{\sqrt {\theta )))+J_{x+{\frac {1}{2))}(2{\sqrt {\theta )))J_{y+{\frac {1}{2))}(2{\sqrt {\theta )))}$$

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).^[6]

Uniform spanning trees

Let G be a finite, undirected, connected graph, with edge set E. Define I^e:E → ℓ²(E) as follows: first choose some arbitrary set of orientations for the edges E, and for each resulting, oriented edge e, define I^e to be the projection of a unit flow along e onto the subspace of ℓ²(E) spanned by star flows.^[9] Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

${\displaystyle K(e,f)=\langle I^{e},I^{f}\rangle ,\quad e,f\in E}$.^[5]