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]

## Contents

* 1 Definition
* 2 Properties
* 2.1 Existence
* 2.2 Uniqueness
* 3 Examples
* 3.1 Gaussian unitary ensemble
* 3.2 Poissonized Plancherel measure
* 3.3 Uniform spanning trees
* 4 References

## 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, ..., xn ∈ Λ.[5]

## Properties

### 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: [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 ρk 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 Ie:E → ℓ2(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 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

1. ↑ Vershik, Anatoly M. (2003). Asymptotic combinatorics with applications to mathematical physics a European mathematical summer school held at the Euler Institute, St. Petersburg, Russia, July 9-20, 2001. Berlin [etc.]: Springer. p. 151. ISBN 978-3-540-44890-7.
2. ↑ Miyoshi, Naoto; Shirai, Tomoyuki (2016). "A Cellular Network Model with Ginibre Configured Base Stations". Advances in Applied Probability 46 (3): 832–845. doi:10.1239/aap/1409319562. ISSN 0001-8678.
3. ↑ Torrisi, Giovanni Luca; Leonardi, Emilio (2014). "Large Deviations of the Interference in the Ginibre Network Model". Stochastic Systems 4 (1): 173–205. doi:10.1287/13-SSY109. ISSN 1946-5238. https://iris.polito.it/bitstream/11583/2525885/2/SSY-2013-109.pdf.
4. ↑ N. Deng, W. Zhou, and M. Haenggi. The Ginibre point process as a model for wireless networks with repulsion. IEEE Transactions on Wireless Communications, vol. 14, pp. 107-121, Jan. 2015.
5. ↑ 5.0 5.1 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.
6. ↑ 6.0 6.1 6.2 A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975.
7. ↑ B. Valko. Random matrices, lectures 14–15. Course lecture notes, University of Wisconsin-Madison.
8. ↑ A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via arXiv:math/9905032.
9. ↑ Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current version available at http://mypage.iu.edu/~rdlyons/

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Determinantal point process. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Determinantal_point_process&oldid=2781620"