The poissonized Plancherel measure on integer partition (and therefore on Young diagramss) 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 + 1⁄2 with the discrete Bessel kernel, given by:
where
For J the Bessel function of the first kind, and θ the mean used in poissonization.[8]
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).[6]
Uniform spanning treesedit
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.[9] Then the uniformly random spanning tree of G is a determinantal point process on E, with kernel
^Vershik, Anatoly M. (2003). Asymptotic combinatorics with applications to mathematical physics a European mathematical summer school held at the Euler Institute, St. Petersburg, Russia, July 9-20, 2001. Berlin [etc.]: Springer. p. 151. ISBN978-3-540-44890-7.
^Miyoshi, Naoto; Shirai, Tomoyuki (2016). "A Cellular Network Model with Ginibre Configured Base Stations". Advances in Applied Probability. 46 (3): 832–845. doi:10.1239/aap/1409319562. ISSN 0001-8678.
^Torrisi, Giovanni Luca; Leonardi, Emilio (2014). "Large Deviations of the Interference in the Ginibre Network Model" (PDF). Stochastic Systems. 4 (1): 173–205. doi:10.1287/13-SSY109. ISSN 1946-5238.
^N. Deng, W. Zhou, and M. Haenggi. The Ginibre point process as a model for wireless networks with repulsion. IEEE Transactions on Wireless Communications, vol. 14, pp. 107-121, Jan. 2015.
^ ab 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.
^ abcA. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975.
^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 arXiv: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/
January 01, 1970
determinantal, point, process, mathematics, determinantal, point, process, stochastic, point, process, probability, distribution, which, characterized, determinant, some, function, such, processes, arise, important, tools, random, matrix, theory, combinatorics. 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 1 and wireless network modeling 2 3 4 Contents 1 Definition 2 Properties 2 1 Existence 2 2 Uniqueness 3 Examples 3 1 Gaussian unitary ensemble 3 2 Poissonized Plancherel measure 3 3 Uniform spanning trees 4 ReferencesDefinition editLet L displaystyle Lambda nbsp be a locally compact Polish space and m displaystyle mu nbsp be a Radon measure on L displaystyle Lambda nbsp Also consider a measurable function K L 2 C displaystyle K Lambda 2 to mathbb C nbsp We say that X displaystyle X nbsp is a determinantal point process on L displaystyle Lambda nbsp with kernel K displaystyle K nbsp if it is a simple point process on L displaystyle Lambda nbsp with a joint intensity or correlation function which is the density of its factorial moment measure given by r n x 1 x n det K x i x j 1 i j n displaystyle rho n x 1 ldots x n det K x i x j 1 leq i j leq n nbsp for every n 1 and x1 xn L 5 Properties editExistence edit The following two conditions are necessary and sufficient for the existence of a determinantal random point process with intensities rk Symmetry rk is invariant under action of the symmetric group Sk Thus r k x s 1 x s k r k x 1 x k s S k k displaystyle rho k x sigma 1 ldots x sigma k rho k x 1 ldots x k quad forall sigma in S k k nbsp Positivity For any N and any collection of measurable bounded functions f k L k R displaystyle varphi k Lambda k to mathbb R nbsp k 1 N with compact support If f 0 k 1 N i 1 i k f k x i 1 x i k 0 for all k x i i 1 k displaystyle varphi 0 sum k 1 N sum i 1 neq cdots neq i k varphi k x i 1 ldots x i k geq 0 text for all k x i i 1 k nbsp Then 6 f 0 k 1 N L k f k x 1 x k r k x 1 x k d x 1 d x k 0 for all k x i i 1 k displaystyle varphi 0 sum k 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 geq 0 text for all k x i i 1 k nbsp Uniqueness edit A sufficient condition for the uniqueness of a determinantal random process with joint intensities rk is k 0 1 k A k r k x 1 x k d x 1 d x k 1 k displaystyle 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 nbsp for every bounded Borel A L 6 Examples editGaussian unitary ensemble edit Main article 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 R displaystyle mathbb R nbsp with kernel K m x y k 0 m 1 ps k x ps k y displaystyle K m x y sum k 0 m 1 psi k x psi k y nbsp where ps k x displaystyle psi k x nbsp is the k displaystyle k nbsp th oscillator wave function defined byps k x 1 2 n n H k x e x 2 4 displaystyle psi k x frac 1 sqrt sqrt 2n n H k x e x 2 4 nbsp and H k x displaystyle H k x nbsp is the k displaystyle k nbsp th Hermite polynomial 7 Poissonized Plancherel measure edit The poissonized Plancherel measure on integer partition and therefore on Young diagramss 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 displaystyle mathbb Z nbsp 1 2 with the discrete Bessel kernel given by K x y 8 k x y x y if x y gt 0 8 k x y x y if x y lt 0 displaystyle K x y begin cases sqrt theta dfrac k x y x y amp text if xy gt 0 12pt sqrt theta dfrac k x y x y amp text if xy lt 0 end cases nbsp where k x y J x 1 2 2 8 J y 1 2 2 8 J x 1 2 2 8 J y 1 2 2 8 displaystyle 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 nbsp k x y J x 1 2 2 8 J y 1 2 2 8 J x 1 2 2 8 J y 1 2 2 8 displaystyle 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 nbsp For J the Bessel function of the first kind and 8 the mean used in poissonization 8 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 6 Uniform spanning trees edit 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 9 Then the uniformly random spanning tree of G is a determinantal point process on E with kernel K e f I e I f e f E displaystyle K e f langle I e I f rangle quad e f in E nbsp 5 References edit Vershik Anatoly M 2003 Asymptotic combinatorics with applications to mathematical physics a European mathematical summer school held at the Euler Institute St Petersburg Russia July 9 20 2001 Berlin etc Springer p 151 ISBN 978 3 540 44890 7 Miyoshi Naoto Shirai Tomoyuki 2016 A Cellular Network Model with Ginibre Configured Base Stations Advances in Applied Probability 46 3 832 845 doi 10 1239 aap 1409319562 ISSN 0001 8678 Torrisi Giovanni Luca Leonardi Emilio 2014 Large Deviations of the Interference in the Ginibre Network Model PDF Stochastic Systems 4 1 173 205 doi 10 1287 13 SSY109 ISSN 1946 5238 N Deng W Zhou and M Haenggi The Ginibre point process as a model for wireless networks with repulsion IEEE Transactions on Wireless Communications vol 14 pp 107 121 Jan 2015 a b Hough J B Krishnapur M Peres Y and Virag B Zeros of Gaussian analytic functions and determinantal point processes University Lecture Series 51 American Mathematical Society Providence RI 2009 a b c A Soshnikov Determinantal random point fields Russian Math Surveys 2000 55 5 923 975 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 arXiv 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 Retrieved from https en wikipedia org w index php title Determinantal point process amp oldid 1211655470, wikipedia, wiki, book, books, library,