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]

Existence edit

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 edit

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

Gaussian unitary ensemble edit

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

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

Poissonized Plancherel measure edit

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:

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 edit

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]