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.

Definition

Let $\Lambda$ be a locally compact Polish space and $\mu$ be a Radon measure on $\Lambda$. Also, consider a measurable function K:Λ² → ℂ.

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

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 $\psi_k(x)$ is the $k$th 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 $k$th 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 + 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. 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. 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/