# 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 ${\displaystyle \Lambda }$ be a locally compact Polish space and ${\displaystyle \mu }$ be a Radon measure on ${\displaystyle \Lambda }$. Also, consider a measurable function K2 → ℂ.

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

${\displaystyle \rho _{n}(x_{1},\ldots ,x_{n})=\det[K(x_{i},x_{j})]_{1\leq i,j\leq n}}$

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.

${\displaystyle \rho _{k}(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\rho _{k}(x_{1},\ldots ,x_{k})\quad \forall \sigma \in S_{k},k}$
• Positivity: For any N, and any collection of measurable, bounded functions φk:Λk → ℝ, k = 1,. . . ,N with compact support:
If
${\displaystyle \quad \varphi _{0}+\sum _{k=1}^{N}\sum _{i_{1}\neq \cdots \neq i_{k}}\varphi _{k}(x_{i_{1}}\ldots x_{i_{k}})\geq 0{\text{ for all }}k,(x_{i})_{i=1}^{k}}$
Then
${\displaystyle \quad \varphi _{0}+\sum _{k=1}^{N}\int _{\Lambda ^{k}}\varphi _{k}(x_{1},\ldots ,x_{k})\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d}}x_{1}\cdots {\textrm {d}}x_{k}\geq 0{\text{ for all }}k,(x_{i})_{i=1}^{k}}$ [6]

### Uniqueness

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

${\displaystyle \sum _{k=0}^{\infty }\left({\frac {1}{k!}}\int _{A^{k}}\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d}}x_{1}\cdots {\textrm {d}}x_{k}\right)^{-{\frac {1}{k}}}=\infty }$

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 ${\displaystyle \mathbb {R} }$ with kernel

${\displaystyle K_{m}(x,y)=\sum _{k=0}^{m-1}\psi _{k}(x)\psi _{k}(y)}$

where ${\displaystyle \psi _{k}(x)}$ is the ${\displaystyle k}$th oscillator wave function defined by

${\displaystyle \psi _{k}(x)={\frac {1}{\sqrt {{\sqrt {2n}}n!}}}H_{k}(x)e^{-x^{2}/4}}$

and ${\displaystyle H_{k}(x)}$ is the ${\displaystyle k}$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 ℤ[clarification needed] + 12 with the discrete Bessel kernel, given by:

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

where

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

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 → 2(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 2(E) spanned by star flows.[9] Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

${\displaystyle K(e,f)=\langle I^{e},I^{f}\rangle ,\quad e,f\in E}$.[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" (PDF). Stochastic Systems. 4 (1): 173–205. doi:10.1287/13-SSY109. ISSN 1946-5238.
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. ^ 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.
6. ^ a b c A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975.
7. ^
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/