 # Determinantal point process

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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 Λ be a locally compact Polish space and μ be a Radon measure on Λ . Also, consider a measurable function K2 → ℂ.

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

ρ n ( x 1 , , x n ) = det ( K ( x i , x j ) 1 i , j 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:
• Positivity: For any N, and any collection of measurable, bounded functions φk:Λk → ℝ, k = 1,. . . ,N with compact support:
• If Then

## Uniqueness

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

k = 0 ( 1 k ! A k ρ k ( x 1 , , x k ) d x 1 d x k ) 1 k =

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 R with kernel

K m ( x , y ) = k = 0 m 1 ψ k ( x ) ψ k ( y )

where ψ k ( x ) is the k th oscillator wave function defined by

ψ k ( x ) = 1 2 n 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 + 12 with the discrete Bessel kernel, given by:

K ( x , y ) = { θ k + ( | x | , | y | ) | x | | y | if  x y > 0 , θ k ( | x | , | y | ) x y if  x y < 0 ,

where

k + ( x , y ) = J x 1 2 ( 2 θ ) J y + 1 2 ( 2 θ ) J x + 1 2 ( 2 θ ) J y 1 2 ( 2 θ ) , k ( x , y ) = J x 1 2 ( 2 θ ) J y 1 2 ( 2 θ ) + J x + 1 2 ( 2 θ ) J y + 1 2 ( 2 θ )

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 ) = I e , I f , e , f E .

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