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, and physics.^{[citation needed]}

Definition

Let Λ be a locally compact Polish space and μ be a Radon measure on Λ. Also, consider a measurable function K:Λ² → 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 joint intensities given by

$\rho_n(x_1,\ldots,x_n) = \det(K(x_i,x_j)_{1 \le i,j \le n})$

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

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:

$\rho_k(x_{\sigma(1)},\ldots,x_{\sigma(k)}) = \rho_k(x_1,\ldots,x_k)\quad \forall \sigma \in S_k, k$

• Positivity: For any N, and any collection of measurable, bounded functions φ_k:Λ^k → ℝ, k = 1,. . . ,N with compact support:

If

$\quad \varphi_0 + \sum_{k=1}^N \sum_{i_1 \neq \cdots \neq i_k } \varphi_k(x_{i_1} \ldots x_{i_k})\ge 0 \text{ for all }k,(x_i)_{i = 1}^k$

Then

$\quad \varphi_0 + \sum_{i=1}^N \int_{\Lambda^k} \varphi_k(x_1, \ldots, x_k)\rho_k(x_1,\ldots,x_k)\,\textrm{d}x_1\cdots\textrm{d}x_k \ge0 \text{ for all } k, (x_i)_{i = 1}^k$ ^[2]

Uniqueness

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

$\sum_{k = 0}^\infty \left( \frac{1}{k!} \int_{A^k} \rho_k(x_1,\ldots,x_k) \, \textrm{d}x_1\cdots\textrm{d}x_k \right)^{-\frac{1}{k}} = \infty$

for every bounded Borel A ⊆ Λ.^[2]

Examples

Gaussian unitary ensemble

Main article: 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 $\mathbb{R}$ with kernel

$K_m(x,y) = \sum_{k=0}^{m-1} \psi_k(x) \psi_k(y)$

where ψ_k(x) is the kth oscillator wave function defined by

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

and H_k(x) is the kth Hermite polynomial. ^[3]

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^{[clarification needed]} + ¹⁄₂ with the discrete Bessel kernel, given by:

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

where

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

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

For J the Bessel function of the first kind, and θ the mean used in poissonization.^[4]

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).^[2]

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.^[5] Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

$K(e,f) = \langle I^e,I^f \rangle ,\quad e,f \in E$.^[1]

References

1. ^ ^{a b} Hough, J. B., Krishnapur, M., Peres, Y., and Virág, B., Zeros of Gaussian analytic functions and determinantal point processes. University Lecture Series, 51. American Mathematical Society, Providence, RI, 2009.
2. ^ ^{a b c} A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975.
3. ^ B. Valko. Random matrices, lectures 14–15. Course lecture notes, University of Wisconsin-Madison.
4. ^ A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via http://xxx.lanl.gov/abs/math/9905032.
5. ^ Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current version available at http://mypage.iu.edu/~rdlyons/