Determinantal point process

Consider some positively charged particles confined in a 1-dimensional box $[-1,\; +1]$. Due to electrostatic repulsion, the locations of the charged particles are negatively correlated. That is, if one particle is in a small segment $[x,\; x\; +\; \backslash delta\; x]$, then that makes the other particles less likely to be in the same set. The strength of repulsion between two particles at locations $x,\; x\text{'}$ can be characterized by a function $K(x,\; x\text{'})$.

Let $\backslash Lambda$ be a locally compact Polish space and $\backslash mu$ be a Radon measure on $\backslash Lambda$. In most concrete applications, these are Euclidean space $\backslash R^n$ with its Lebesgue measure. A kernel function is a measurable function $K:\; \backslash Lambda^2\; \backslash to\; \backslash mathbb\{C\}$.

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

:$\backslash rho\_n(x\_1,\backslash ldots,x\_n)\; =\; \backslash det[K(x\_i,x\_j)]\_\{1\; \backslash le\; i,j\; \backslash le\; n\}$

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

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: $$\backslash rho\_k(x\_\{\backslash sigma(1)\},\; \backslash ldots,\; x\_\{\backslash sigma(k)\})\; =\; \backslash rho\_k(x\_1,\; \backslash ldots,\; x\_k)\backslash quad\; \backslash forall\; \backslash sigma\; \backslash in\; S\_k,\; k$$
• Positivity: For any N, and any collection of measurable, bounded functions {{nowrap|$\backslash varphi\_k\; :\; \backslash Lambda^k\; \backslash to\; \backslash mathbb\{R\}$,}} k = 1, ..., N with compact support:{{pb}} If $$\backslash varphi\_0\; +\; \backslash sum\_\{k=1\}^N\; \backslash sum\_\{i\_1\; \backslash neq\; \backslash cdots\; \backslash neq\; i\_k\; \}\; \backslash varphi\_k(x\_\{i\_1\}\; \backslash ldots\; x\_\{i\_k\})\backslash ge\; 0\; \backslash text\{\; for\; all\; \}k,(x\_i)\_\{i\; =\; 1\}^k$$ Then A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975. $$\backslash varphi\_0\; +\; \backslash sum\_\{k=1\}^N\; \backslash int\_\{\backslash Lambda^k\}\; \backslash varphi\_k(x\_1,\; \backslash ldots,\; x\_k)\backslash rho\_k(x\_1,\backslash ldots,x\_k)\backslash ,\backslash textrm\{d\}x\_1\backslash cdots\backslash textrm\{d\}x\_k\; \backslash ge0\; \backslash text\{\; for\; all\; \}\; k,\; (x\_i)\_\{i\; =\; 1\}^k$$

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

$$\backslash sum\_\{k\; =\; 0\}^\backslash infty\; \backslash left(\; \backslash frac\{1\}\{k!\}\; \backslash int\_\{A^k\}\; \backslash rho\_k(x\_1,\backslash ldots,x\_k)\; \backslash ,\; \backslash textrm\{d\}x\_1\backslash cdots\backslash textrm\{d\}x\_k\; \backslash right)^\{-\backslash frac\{1\}\{k\}\}\; =\; \backslash infty$$

for every bounded Borel {{nowrap|A ⊆ Λ.}}

{{Main|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 $\backslash mathbb\{R\}$ with kernel

:$K\_m(x,y)\; =\; \backslash sum\_\{k=0\}^\{m-1\}\; \backslash psi\_k(x)\; \backslash psi\_k(y)$

where $\backslash psi\_k(x)$ 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.

B. Valko. [http://www.math.wisc.edu/%7Evalko/courses/833/lec_14_15.pdf Random matrices, lectures 14–15]. [http://www.math.wisc.edu/%7Evalko/courses/833/833.html Course lecture notes, University of Wisconsin-Madison].

{{main|Airy process}}

The Airy process is governed by the so called extended Airy kernel which is a generalization of the Airy kernel function$$K^\{\backslash mathrm\{Ai\}\}(x,\; y)=\backslash frac\{\backslash operatorname\{Ai\}(x)\; \backslash operatorname\{Ai\}^\{\backslash prime\}(y)-\backslash operatorname\{Ai\}(y)\; \backslash operatorname\{Ai\}^\{\backslash prime\}(x)\}\{x-y\}$$where $\backslash operatorname\{Ai\}$ is the Airy function. This process arises from rescaled eigenvalues near the spectral edge of the Gaussian Unitary Ensemble.{{Cite journal |last=Tracy |first=Craig A. |last2=Widom |first2=Harold |date=January 1994 |title=Level-spacing distributions and the Airy kernel |url=http://link.springer.com/10.1007/BF02100489 |journal=Communications in Mathematical Physics |language=en |volume=159 |issue=1 |pages=151–174 |doi=10.1007/BF02100489 |issn=0010-3616|arxiv=hep-th/9211141 }}

The poissonized Plancherel measure on integer partition (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 $\backslash mathbb\{Z\}$ + {{frac|2|}} with the discrete Bessel kernel, given by:

$$K(x,y)\; =$$

\begin{cases}

\sqrt{\theta} \, \dfrac{k_+(|x|,|y|)}

\sqrt{\theta} \, \dfrac{k_-(|x|,|y|)}{x-y} & \text{if } xy <0,

\end{cases}

where

$$k\_+(x,y)\; =\; J\_\{x-\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})J\_\{y+\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})\; -\; J\_\{x+\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})J\_\{y-\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\}),$$

$$k\_-(x,y)\; =\; J\_\{x-\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})J\_\{y-\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})\; +\; J\_\{x+\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})J\_\{y+\backslash frac\{1\}\{2\}\}(2\backslash sqrt\{\backslash theta\})$$

For J the Bessel function of the first kind, and θ the mean used in poissonization.A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via {{arXiv|math/9905032}}.

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).

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.Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current

version available at http://mypage.iu.edu/~rdlyons/ Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

:$K(e,f)\; =\; \backslash langle\; I^e,I^f\; \backslash rangle\; ,\backslash quad\; e,f\; \backslash in\; E$.

{{Reflist}}

• {{cite book |last=Johansson |first=Kurt |title=Course 1 - Random matrices and determinantal processes |publisher=Elsevier |year=2006 |series=Les Houches |volume=83 |chapter= |doi=10.1016/s0924-8099(06)80038-7 |issn=0924-8099}}

Category:Point processes