Inmathematics, adeterminantal point process is astochasticpoint process, theprobability distribution of which is characterized as adeterminant of some function. They are suited for modelling global negative correlations, and for efficient algorithms of sampling, marginalization, conditioning, and other inference tasks. Such processes arise as important tools inrandom matrix theory,combinatorics,physics,^[1]machine learning,^[2] and wireless network modeling.^[3][4][5]

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

Let${\displaystyle \mathrm{\Lambda }}$ be alocally compactPolish space and${\displaystyle \mu }$ be aRadon measure on${\displaystyle \mathrm{\Lambda }}$. In most concrete applications, these areEuclidean space${\displaystyle {\mathbb{R}}^n}$ with its Lebesgue measure. Akernel function is ameasurable function${\displaystyle K:{\mathrm{\Lambda }}^2\to \mathbb{C}}$.

We say that${\displaystyle X}$ is adeterminantal point process on${\displaystyle \mathrm{\Lambda }}$ with kernel${\displaystyle K}$ if it is a simplepoint process on${\displaystyle \mathrm{\Lambda }}$ with ajoint intensity orcorrelation function (which is the density of itsfactorial moment measure) given by

${\displaystyle {\rho }_n(x_1,\dots ,x_n)=det[K(x_i,x_j){]}_{1\le i,j\le n}}$

for everyn ≥ 1 andx₁, ...,x_n ∈ Λ.^[6]

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 thesymmetric groupS_k. Thus:$${\displaystyle {\rho }_k(x_{\sigma (1)},\dots ,x_{\sigma (k)})={\rho }_k(x_1,\dots ,x_k)\mathrm{\forall }\sigma \in S_k,k}$$
• Positivity: For anyN, and any collection of measurable, bounded functions${\displaystyle {\phi }_k:{\mathrm{\Lambda }}^k\to \mathbb{R}}$,k = 1, ...,N withcompact support:
  If$${\displaystyle {\phi }_0+\sum _{k=1}^N\sum _{i_1\ne \cdots \ne i_k}{\phi }_k(x_{i_1}\dots x_{i_k})\ge 0\text{ for all }k,(x_i{)}_{i=1}^k}$$ Then^[7]$${\displaystyle {\phi }_0+\sum _{k=1}^N{\int }_{{\mathrm{\Lambda }}^k}{\phi }_k(x_1,\dots ,x_k){\rho }_k(x_1,\dots ,x_k)\text{d}{x}_1\cdots \text{d}{x}_k\ge 0\text{ for all }k,(x_i{)}_{i=1}^k}$$

A sufficient condition for the uniqueness of a determinantal random process with joint intensitiesρ_k is$${\displaystyle \sum _{k=0}^{\mathrm{\infty }}{\left(\frac{1}{k!}{\int }_{A^k}{\rho }_k(x_1,\dots ,x_k)\text{d}{x}_1\cdots \text{d}{x}_k\right)}^{-\frac{1}{k}}=\mathrm{\infty }}$$for every bounded BorelA ⊆ Λ.^[7]

The eigenvalues of a randomm × m Hermitian matrix drawn from theGaussian 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.^[8]

TheAiry process is governed by the so calledextended Airy kernel which is a generalization of the Airy kernel function$${\displaystyle K^{\mathrm{A}\mathrm{i}}(x,y)=\frac{\mathrm{Ai}(x){\mathrm{Ai}}^{\mathrm{\prime }}(y)-\mathrm{Ai}(y){\mathrm{Ai}}^{\mathrm{\prime }}(x)}{x-y}}$$where${\displaystyle \mathrm{Ai}}$ is theAiry function. This process arises from rescaled eigenvalues near the spectral edge of theGaussian Unitary Ensemble.^[9]

The poissonized Plancherel measure oninteger partition (and therefore onYoung diagrams) plays an important role in the study of thelongest 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${\displaystyle \mathbb{Z}}$ +1⁄2 with the discrete Bessel kernel, given by:

where$${\displaystyle k_{+}(x,y)=J_{x-\frac{1}{2}}(2\sqrt{\theta })J_{y+\frac{1}{2}}(2\sqrt{\theta })-J_{x+\frac{1}{2}}(2\sqrt{\theta })J_{y-\frac{1}{2}}(2\sqrt{\theta }),}$$$${\displaystyle k_{-}(x,y)=J_{x-\frac{1}{2}}(2\sqrt{\theta })J_{y-\frac{1}{2}}(2\sqrt{\theta })+J_{x+\frac{1}{2}}(2\sqrt{\theta })J_{y+\frac{1}{2}}(2\sqrt{\theta })}$$ForJ theBessel function of the first kind, and θ the mean used in poissonization.^[10]

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).^[7]

Let G be a finite, undirected, connectedgraph, with edge setE. DefineI^e:E → ℓ²(E) as follows: first choose some arbitrary set of orientations for the edges E, and for each resulting, oriented edgee, defineI^e to be the projection of a unit flow alonge onto the subspace ofℓ²(E) spanned by star flows.^[11] Then the uniformly randomspanning tree of G is a determinantal point process onE, with kernel

${\displaystyle K(e,f)=⟨I^e,I^f⟩,e,f\in E}$.^[6]

• Johansson, Kurt (2006).Course 1 - Random matrices and determinantal processes. Les Houches. Vol. 83. Elsevier.doi:10.1016/s0924-8099(06)80038-7.ISSN 0924-8099.

• Point processes

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