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, and wireless network modeling. 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, and wireless network modeling. Let Λ {displaystyle Lambda } be a locally compact Polish space and μ {displaystyle mu } be a Radon measure on Λ {displaystyle Lambda } . Also, consider a measurable function K:Λ2 → ℂ. We say that X {displaystyle X} is a determinantal point process on Λ {displaystyle Lambda } with kernel K {displaystyle K} if it is a simple point process on Λ {displaystyle Lambda } with a joint intensity or correlation function (which is the density of its factorial moment measure) given by for every n ≥ 1 and x1, . . . , xn ∈ Λ. The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities ρk. A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρk is for every bounded Borel A ⊆ Λ. The eigenvalues of a random m × m Hermitian matrix drawn from the Gaussian unitary ensemble (GUE) form a determinantal point process on R {displaystyle mathbb {R} } with kernel where ψ k ( x ) {displaystyle psi _{k}(x)} is the k {displaystyle k} th oscillator wave function defined by