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. Let be a locally compact Polish space and be a Radon measure on . Also, consider a measurable function . We say that is a determinantal point process on with kernel if it is a simple point process on 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. Symmetry: ρk is invariant under action of the symmetric group Sk. Thus: Positivity: For any N, and any collection of measurable, bounded functions , k = 1, ..., N with compact support: If Then A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρk is for every bounded Borel A ⊆ Λ. 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 with kernel where is the th oscillator wave function defined by and is the th Hermite polynomial. 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 + with the discrete Bessel kernel, given by: where For J the Bessel function of the first kind, and θ the mean used in poissonization. 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.