Determinantal point process

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 ${\displaystyle x,x'}$  can be characterized by a function ${\displaystyle K(x,x')}$ .

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

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, ..., 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 the symmetric group S_k. Thus: $${\displaystyle \rho _{k}(x_{\sigma (1)},\ldots ,x_{\sigma (k)})=\rho _{k}(x_{1},\ldots ,x_{k})\quad \forall \sigma \in S_{k},k}$$ 
• Positivity: For any N, and any collection of measurable, bounded functions ${\displaystyle \varphi _{k}:\Lambda ^{k}\to \mathbb {R} }$ , k = 1, ..., N with compact support:
  If $${\displaystyle \varphi _{0}+\sum _{k=1}^{N}\sum _{i_{1}\neq \cdots \neq i_{k}}\varphi _{k}(x_{i_{1}}\ldots x_{i_{k}})\geq 0{\text{ for all }}k,(x_{i})_{i=1}^{k}}$$  Then ^[7] $${\displaystyle \varphi _{0}+\sum _{k=1}^{N}\int _{\Lambda ^{k}}\varphi _{k}(x_{1},\ldots ,x_{k})\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d}}x_{1}\cdots {\textrm {d}}x_{k}\geq 0{\text{ for all }}k,(x_{i})_{i=1}^{k}}$$ 

A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρ_k is $${\displaystyle \sum _{k=0}^{\infty }\left({\frac {1}{k!}}\int _{A^{k}}\rho _{k}(x_{1},\ldots ,x_{k})\,{\textrm {d}}x_{1}\cdots {\textrm {d}}x_{k}\right)^{-{\frac {1}{k}}}=\infty }$$  for every bounded Borel A ⊆ Λ.^[7]

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. ^[8]

The Airy process has kernel function$${\displaystyle K^{\mathrm {Ai} }(x,y)={\frac {\operatorname {Ai} (x)\operatorname {Ai} ^{\prime }(y)-\operatorname {Ai} (y)\operatorname {Ai} ^{\prime }(x)}{x-y}}}$$ where ${\displaystyle \operatorname {Ai} }$  is the Airy function. This process arises from rescaled eigenvalues near the spectral edge of the Gaussian Unitary Ensemble. It was introduced in 1992.^[9]

The poissonized Plancherel measure on integer partition (and therefore on Young diagramss) 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 ${\displaystyle \mathbb {Z} }$  + 1⁄2 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.^[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, 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.^[11] Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

${\displaystyle K(e,f)=\langle I^{e},I^{f}\rangle ,\quad e,f\in E}$ .^[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.