Determinantal point process

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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.[citation needed]

Definition[edit]

Let be a locally compact Polish space and be a Radon measure on . Also, consider a measurable function K2 → ℂ.

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.[1]

Properties[edit]

Existence[edit]

The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities ρk.

  • Positivity: For any N, and any collection of measurable, bounded functions φk:Λk → ℝ, k = 1,. . . ,N with compact support:
If
Then
[2]

Uniqueness[edit]

A sufficient condition for the uniqueness of a determinantal random process with joint intensities ρk is

for every bounded Borel A ⊆ Λ.[2]

Examples[edit]

Gaussian unitary ensemble[edit]

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. [3]

Poissonized Plancherel measure[edit]

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] + 12 with the discrete Bessel kernel, given by:

where

For J the Bessel function of the first kind, and θ the mean used in poissonization.[4]

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).[2]

Uniform spanning trees[edit]

Let G be a finite, undirected, connected graph, with edge set E. Define Ie:E → 2(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 2(E) spanned by star flows.[5] Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel

.[1]

References[edit]

  1. ^ a b Hough, J. B., Krishnapur, M., Peres, Y., and Virág, B., Zeros of Gaussian analytic functions and determinantal point processes. University Lecture Series, 51. American Mathematical Society, Providence, RI, 2009.
  2. ^ a b c A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975.
  3. ^ B. Valko. Random matrices, lectures 14–15. Course lecture notes, University of Wisconsin-Madison.
  4. ^ A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via http://xxx.lanl.gov/abs/math/9905032.
  5. ^ Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current version available at http://mypage.iu.edu/~rdlyons/