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| In [[mathematics]], a '''determinantal point process''' is a [[stochastic process|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]].{{cn|date=October 2011}}
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| ==Definition==
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| Let <math>\Lambda</math> be a [[locally compact]] [[Polish space]] and <math>\mu</math> be a Radon measure on <math>\Lambda</math>. Also, consider a measurable function ''K'':Λ<sup>2</sup> → ℂ.
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| We say that <math>X</math> is a '''determinantal point process''' on <math>\Lambda</math> with kernel <math>K</math> if it is a simple point process on <math>\Lambda</math> with a [[Factorial_moment_measure#Factorial_moment_density|joint intensity]] or ''correlation function'' (which is the derivative of its [[factorial moment measure]]) given by
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| :<math> \rho_n(x_1,\ldots,x_n) = \det(K(x_i,x_j)_{1 \le i,j \le n}) </math>
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| for every ''n'' ≥ 1 and ''x''<sub>1</sub>, . . . , ''x''<sub>''n''</sub> ∈ Λ.<ref name=GAF> 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.</ref>
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| ==Properties==
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| ===Existence===
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| The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities ρ<sub>k</sub>. | |
| * Symmetry: ''ρ''<sub>''k''</sub> is invariant under action of the [[symmetric group]] ''S''<sub>''k''</sub>. Thus:
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| ::<math>\rho_k(x_{\sigma(1)},\ldots,x_{\sigma(k)}) = \rho_k(x_1,\ldots,x_k)\quad \forall \sigma \in S_k, k</math>
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| * Positivity: For any ''N'', and any collection of measurable, bounded functions ''φ''<sub>''k''</sub>:''Λ''<sup>''k''</sup> → ℝ, ''k'' = ''1'',. . . ,''N'' with compact support:
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| :If
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| ::<math>\quad \varphi_0 + \sum_{k=1}^N \sum_{i_1 \neq \cdots \neq i_k } \varphi_k(x_{i_1} \ldots x_{i_k})\ge 0 \text{ for all }k,(x_i)_{i = 1}^k </math>
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| :Then
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| ::<math>\quad \varphi_0 + \sum_{i=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 \ge0 \text{ for all } k, (x_i)_{i = 1}^k </math> <ref name=Soshniko>A. Soshnikov, Determinantal random point fields. ''Russian Math. Surveys'', 2000, 55 (5), 923–975.</ref>
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| ===Uniqueness===
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| A sufficient condition for the uniqueness of a determinantal random process with joint intensities ''ρ''<sub>''k''</sub> is
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| : <math>\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</math>
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| for every bounded Borel ''A'' ⊆ ''Λ''.<ref name=Soshniko/>
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| ==Examples==
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| ===Gaussian unitary ensemble===
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| {{Main|Gaussian unitary ensemble}}
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| The eigenvalues of a random ''m'' × ''m'' Hermitian matrix drawn from the [[Gaussian unitary ensemble]] (GUE) form a determinantal point process on <math>\mathbb{R}</math> with kernel
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| :<math>K_m(x,y) = \sum_{k=0}^{m-1} \psi_k(x) \psi_k(y)</math>
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| where <math>\psi_k(x)</math> is the <math>k</math>th oscillator wave function defined by
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| :<math>
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| \psi_k(x)= \frac{1}{\sqrt{\sqrt{2n}n!}}H_k(x) e^{-x^2/4}
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| </math>
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| and <math>H_k(x)</math> is the <math>k</math>th [[Hermite polynomials | Hermite polynomial]].
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| <ref name=Valko>B. Valko. [http://www.math.wisc.edu/%7Evalko/courses/833/lec_14_15.pdf Random matrices, lectures 14–15]. [http://www.math.wisc.edu/%7Evalko/courses/833/833.html Course lecture notes, University of Wisconsin-Madison].</ref>
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| ===Poissonized Plancherel measure===
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| The poissonized Plancherel measure on [[Partition (number theory)|partitions]] of integers (and therefore on [[Young tableaux|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{{clarify|reason=what symbol is this? appears garbage on screen and in editor|date=October 2011}} + {{frac|2|}} with the discrete Bessel kernel, given by:
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| :<math>K(x,y) =
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| \begin{cases}
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| \sqrt{\theta} \, \dfrac{k_+(|x|,|y|)}{|x|-|y|} & \text{if } xy >0,\\[12pt]
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| \sqrt{\theta} \, \dfrac{k_-(|x|,|y|)}{x-y} & \text{if } xy <0,
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| \end{cases} </math>
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| where
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| :<math> 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}), </math>
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| :<math> 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}) </math>
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| For ''J'' the [[Bessel function]] of the first kind, and θ the mean used in poissonization.<ref>A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of Plancherel measures for symmetric groups, available via http://xxx.lanl.gov/abs/math/9905032.</ref>
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| This serves as an example of a well-defined determinantal point process with non-[[Hermitian function|Hermitian]] kernel (although its restriction to the positive and negative semi-axis is Hermitian).<ref name=Soshniko/>
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| ===Uniform spanning trees===
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| Let G be a finite, undirected, connected [[Graph theory|graph]], with edge set ''E''. Define ''I<sup>e</sup>'':''E'' → ''ℓ<sup>2</sup>(E)'' as follows: first choose some arbitrary set of orientations for the edges E, and for each resulting, oriented edge ''e'', define ''I<sup>e</sup>'' to be the projection of a unit flow along ''e'' onto the subspace of ''ℓ<sup>2</sup>(E)'' spanned by star flows.<ref>Lyons, R. with Peres, Y., Probability on Trees and Networks. Cambridge University Press, In preparation. Current
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| version available at http://mypage.iu.edu/~rdlyons/ </ref> Then the uniformly random spanning tree of G is a determinantal point process on ''E'', with kernel
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| :<math>K(e,f) = \langle I^e,I^f \rangle ,\quad e,f \in E</math>.<ref name=GAF/>
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| ==References==
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| {{Reflist}}
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| [[Category:Point processes]]
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The sportfishing fraternity is split across subject employing the lay-in or alternatively take a seat on ideal kayaks concerning hobie cat kayak sail kit take off fishing. historically, paddlers considering wing sportfishing utilised the most common lay-as part of kayaks merely. unfortunately, down dead numerous employ brand new lay on top kayaks while they offer better reliability and are generally nearer to the skin. Lots of paddlers additionally choose go with important fisherman kayaks gold watches which are designed with angling essentials concerning fly day fishing.
Another benefit of soar reef fishing originating from a canoe would it be might non make any sounds. Within the lack of just about any deafening propeller otherwise flipper, paddlers will inaudibly strategy some of the go fishing and take good grab. The greatest advantage of the Hobie cat kayak in fly sport fishing is the fact paddlers are going to go people locations where nothing else fisherman can head.
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