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, and physics.
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 φk:Λk → ℝ, k = 1,. . . ,N with compact support:
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
Poissonized Plancherel measure
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 ℤ[clarification needed] + 1⁄2 with the discrete Bessel kernel, given by:
Uniform spanning trees
Let G be a finite, undirected, connected graph, with edge set E. Define Ie:E → ℓ2(E) as follows: first choose some arbitrary set of orientations for the edges E, and for each resulting, oriented edge e, define Ie to be the projection of a unit flow along e onto the subspace of ℓ2(E) spanned by star flows. Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel
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- B. Valko. Random matrices, lectures 14–15. Course lecture notes, University of Wisconsin-Madison.
- A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via http://xxx.lanl.gov/abs/math/9905032.
- Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current version available at http://mypage.iu.edu/~rdlyons/