Английская Википедия: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>\Lambda</math> be a locally compact Polish space and <math>\mu</math> be a Radon measure on <math>\Lambda</math>. Also, consider a measurable function <math>K: \Lambda^2 \to \mathbb{C}</math>.
We say that <math>X</math> is a determinantal point process on <math>\Lambda</math> with kernel <math>K</math> if it is a simple point process on <math>\Lambda</math> with a joint intensity or correlation function (which is the density of its factorial moment measure) given by
- <math> \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 display="block">\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 Шаблон:Nowrap k = 1, ..., N with compact support:Шаблон:Pb If <math display="block"> \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 display="block"> \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 display="block">\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 Шаблон:Nowrap[6]
Examples
Gaussian unitary ensemble
Шаблон:Main The eigenvalues of a random m × m Hermitian matrix drawn from the Gaussian unitary ensemble (GUE) form a determinantal point process on <math>\mathbb{R}</math> with kernel
- <math>K_m(x,y) = \sum_{k=0}^{m-1} \psi_k(x) \psi_k(y)</math>
where <math>\psi_k(x)</math> is the <math>k</math>th oscillator wave function defined by
- <math>
\psi_k(x)= \frac{1}{\sqrt{\sqrt{2n}n!}}H_k(x) e^{-x^2/4} </math>
and <math>H_k(x)</math> is the <math>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>\mathbb{Z}</math> + Шаблон:Frac with the discrete Bessel kernel, given by:
- <math>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} </math> where
- <math> 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> 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>K(e,f) = \langle I^e,I^f \rangle ,\quad e,f \in E</math>.[5]
References
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ 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.
- ↑ 5,0 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.
- ↑ 6,0 6,1 6,2 A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975.
- ↑ B. Valko. Random matrices, lectures 14–15. Course lecture notes, University of Wisconsin-Madison.
- ↑ A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via Шаблон:ArXiv.
- ↑ Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current version available at http://mypage.iu.edu/~rdlyons/
