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

Definition

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

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

[math]\displaystyle{ \rho_n(x_1,\ldots,x_n) = \det[K(x_i,x_j)]_{1 \le i,j \le n} }[/math]

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: [math]\displaystyle{ \rho_k(x_{\sigma(1)}, \ldots, x_{\sigma(k)}) = \rho_k(x_1, \ldots, x_k)\quad \forall \sigma \in S_k, k }[/math]
• Positivity: For any N, and any collection of measurable, bounded functions [math]\displaystyle{ \varphi_k : \Lambda^k \to \mathbb{R} }[/math], k = 1, ..., N with compact support:
  If [math]\displaystyle{ \varphi_0 + \sum_{k=1}^N \sum_{i_1 \neq \cdots \neq i_k } \varphi_k(x_{i_1} \ldots x_{i_k})\ge 0 \text{ for all }k,(x_i)_{i = 1}^k }[/math] Then ^[6] [math]\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 \ge0 \text{ for all } k, (x_i)_{i = 1}^k }[/math]

Uniqueness

A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρ_k is [math]\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 }[/math] for every bounded Borel A ⊆ Λ.^[6]

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 [math]\displaystyle{ \mathbb{R} }[/math] with kernel

[math]\displaystyle{ K_m(x,y) = \sum_{k=0}^{m-1} \psi_k(x) \psi_k(y) }[/math]

where [math]\displaystyle{ \psi_k(x) }[/math] is the [math]\displaystyle{ k }[/math]th oscillator wave function defined by

[math]\displaystyle{ \psi_k(x)= \frac{1}{\sqrt{\sqrt{2n}n!}}H_k(x) e^{-x^2/4} }[/math]

and [math]\displaystyle{ H_k(x) }[/math] is the [math]\displaystyle{ k }[/math]th Hermite polynomial. ^[7]

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 [math]\displaystyle{ \mathbb{Z} }[/math] + ​¹⁄₂ with the discrete Bessel kernel, given by:

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

where

[math]\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}), }[/math]

[math]\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}) }[/math]

For J the Bessel function of the first kind, and θ the mean used in poissonization.^[8]

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

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

References

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