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 be a locally compact Polish space and be a Radon measure on. Also, consider a measurable function K:Λ2 → ℂ.
We say that is a determinantal point process on with kernel if it is a simple point process on with a joint intensity or correlation function given by
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.
A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρk is
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 form a determinantal point process on with kernel
where is the th oscillator wave function defined by
and is the th Hermite polynomial.

Poissonized Plancherel measure

The poissonized Plancherel measure on partitions of integers 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 + with the discrete Bessel kernel, given by:
where
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.

Uniform spanning trees

Let G be a finite, undirected, connected graph, with edge set E. Define Ie:E → ℓ2 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 spanned by star flows. Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel