A connection between representation of compact groups and some invariant ensembles of hermitian matrices is described. We focus on two types of invariant ensembles which extend the gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal.
Nous décrivons les liens unissant les représentations de groupes compacts et certains ensembles invariants de matrices aléatoires. Cet article porte plus particulièrement sur deux types d'ensembles invariants qui généralisent les ensembles gaussiens ou de Laguerre. Nous les étudions en considérant des convolutions ou des projections de probabilités invariantes sur des orbites adjointes de groupes de Lie compacts. Par approximation semi-classique, ces mesures sont décrites par des produits tensoriels ou des restrictions de représentations. Nous montrons qu'une large classe d'entre elles sont déterminantales.
Keywords: random matrix, determinantal process, interlaced configuration, Gelfand Tsetlin polytope, cristal graph, minor process, rank one perturbation
@article{AIHPB_2010__46_1_209_0, author = {Defosseux, Manon}, title = {Orbit measures, random matrix theory and interlaced determinantal processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {209--249}, publisher = {Gauthier-Villars}, volume = {46}, number = {1}, year = {2010}, doi = {10.1214/09-AIHP314}, mrnumber = {2641777}, zbl = {1216.15024}, language = {en}, url = {http://www.numdam.org/articles/10.1214/09-AIHP314/} }
TY - JOUR AU - Defosseux, Manon TI - Orbit measures, random matrix theory and interlaced determinantal processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2010 SP - 209 EP - 249 VL - 46 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/09-AIHP314/ DO - 10.1214/09-AIHP314 LA - en ID - AIHPB_2010__46_1_209_0 ER -
%0 Journal Article %A Defosseux, Manon %T Orbit measures, random matrix theory and interlaced determinantal processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2010 %P 209-249 %V 46 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/09-AIHP314/ %R 10.1214/09-AIHP314 %G en %F AIHPB_2010__46_1_209_0
Defosseux, Manon. Orbit measures, random matrix theory and interlaced determinantal processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 1, pp. 209-249. doi : 10.1214/09-AIHP314. http://www.numdam.org/articles/10.1214/09-AIHP314/
[1] Nonstandard symmetry classes in mesoscopic normal/superconducting hybrid structures. Phys. Rev. B 55 (1997) 1142-1161.
and .[2] GUEs and queues. Probab. Theory Related Fields 119 (2001) 256-274. | MR | Zbl
.[3] Tensor product multiplicities and convex polytopes in partition space. J. Geom. Phys. 5 (1988) 453-472. | MR | Zbl
and .[4] Le théorème de Pitman. In le groupe quantique SUq(2) et une question de P.A. Meyer. In In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX 61-75. Lecture Notes in Math. 1874. Springer, Berlin, 2006. | MR | Zbl
.[5] Sur les principes de la théorie cinétique des gaz. Ann. Sci. École Norm. Sup. 23 (1906) 9-32. | JFM | Numdam | MR
.[6] Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes. Ann. of Math. (2) 161 (2005) 1319-1422. | MR | Zbl
and .[7] Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055-1080. | MR | Zbl
, , and .[8] Biorthogonal ensembles. Nuclear Phys. B 536 (1999) 704-732. | MR | Zbl
.[9] Level statistics inside the vortex of a superconductor and symplectic random-matrix theory in an external source. Phys. Rev. B 60 (1999) 3589-3602.
, and .[10] Intersection numbers from the antisymmetric Gaussian matrix model. J. High Energy Phys. (2008) 7 050, 19. | MR
and .[11] Wishart process. J. Theoret. Probab. 4 (1991) 725-751. | MR | Zbl
.[12] Network models in class C on arbitrary graphs. Comm. Math. Phys. 258 (2005) 87-102. | MR | Zbl
.[13] Random matrix theory and symmetric spaces. Phys. Rep. 394 (2004) 41-156. | MR
and .[14] The shape of a typical boxed plane partition. New York J. Math. 4 (1998) 137-165 (electronic). | EuDML | MR | Zbl
, and .[15] Representation of Lie groups and random matrices. Trans. Amer. Math. Soc. 361 (2009) 3269-3287. | MR | Zbl
and .[16] Orbit measures and interlaced determinantal point processes. C. R. Math. Acad. Sci. Paris Ser. I 346 (2008) 783-788. | MR | Zbl
.[17] Products of random matrices as they arise in the study of random walks on groups. In Random Matrices and Their Applications (Brunswick, Maine, 1984) 183-195. Contemp. Math. 50. Amer. Math. Soc., Providence, RI, 1986. | MR | Zbl
and .[18] Sums of adjoint orbits. Linear Multilinear Algebra 36 (1993) 79-101. | MR | Zbl
, and .[19] Harmonic analysis and the global exponential map for compact Lie groups. Funct. Anal. Appl. 27 (1993) 25-32. | MR | Zbl
and .[20] The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3 (1962) 1199-1215. | MR | Zbl
.[21] Large deviations for random matrix ensembles in mesoscopic physics. Available at arXiv:math/0610811v2. | MR | Zbl
and .[22] Infinite Dimensional Spherical Analysis. COE Lecture Note 10. Kyushu University, The 21st Century COE Program “DMHF,” Fukuoka, 2008. | MR | Zbl
.[23] Log-gases and random matrices. To appear. Available at http://www.ms.unimelb.edu.au/~matpjf/matpjf.html. | MR | Zbl
.[24] The anti-symmetric GUE minor process. Available at arXiv:math-pr/0804.3293v1. | MR | Zbl
and .[25] Lattice paths proofs for determinantal formulas for symplectic and orthogonal characters. J. Combin. Theory Ser. A 77 (1997) 3-50. | MR | Zbl
and .[26] Young Tableaux. London Mathematical Society Student Text 35. Cambridge Univ. Press, Cambridge, 1997. | MR | Zbl
.[27] Finite dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk. USSR 71 (1981) 275-290.
and .[28] Asymptotic behaviour of watermelons. Preprint, 2003. Available at arXiv:math.PR/0307204.
.[29] Statistical analysis based on a certain multivariate complex Gaussian distribution (An Introduction). Ann. Math. Statist. 34 (1963) 152-177. | MR | Zbl
.[30] Projections of orbits and asymptotic behaviour of multiplicities for compact connected Lie groups. Invent. Math. 67 (1982) 333-356. | MR | Zbl
.[31] Symmetry classes of disordered fermions. Commun. Math. Phys. 257 (2005) 725-771. | MR | Zbl
, and .[32] Groups and Geometric Analysis. Academic Press, New York, 1984. | MR | Zbl
.[33] Random matrices and determinantal processes. Available at arXiv:math-ph/0510038v1. | MR
.[34] Eigenvalues of GUE minors. Electron. J. Probab. 11 (2006) 1342-1371. | MR | Zbl
and .[35] On crystal bases. In Representations of Groups. CMS Conference Proceedings 16 155-197. Amer. Math. Soc., Providence, RI, 1995. | MR | Zbl
.[36] Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165 (1994) 295-345. | MR | Zbl
and .[37] Vicious walk with a wall, noncolliding meanders, chiral and Bogoliubov-de Gennes random matrices. Phys. Rev. E 68 (2003) 1-16.
, , and .[38] Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004) 3058-3085. | MR | Zbl
and .[39] Merits and demerits of the orbit method. Bull. Amer. Math. Soc. (N.S.) 36 (1999) 433-488. | MR | Zbl
.[40] Lectures on the Orbit Method. Graduate Studies in Mathematics 64. Amer. Math. Soc., Providence, RI, 2004. | MR
.[41] Random walks on symmetric spaces and inequalities for matrix spectra. Linear Algebra Appl. 319 (2000) 37-59. | MR | Zbl
.[42] Lie Groups, Beyond an Introduction, 2nd edition. Progress in Mathematics 140. Birkhäuser, Boston, 2002. | MR | Zbl
.[43] Vicious walkers, friendly walkers and Young tableaux. II. With a wall. J. Phys. A 33 (2000) 8835-8866. | MR | Zbl
, and .[44] Distribution laws for the roots of a random antisymmetric hermitian matrix. Nuclear Phys. A 109 (1968) 449-456.
and .[45] Random Matrices, 3rd edition. Pure and Applied Mathematics (Amsterdam) 142. Elsevier-Academic Press, Amsterdam, 2004. | MR | Zbl
.[46] Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras. Comm. Math. Phys. 154 (1993) 215-243. | MR | Zbl
.[47] The birth of random matrix. Moscow Math. J. 6 (2006) 553-566. | MR | Zbl
and .[48] Unitary representations of (G, K)-pairs that are connected with the infinite symmetric group S(∞). Leningrad Math. J. 1 (1990) 983-1014. | MR | Zbl
.[49] Unitary representation of infinite dimensional pairs (G, K) and the formalism of R. Howe. In Representation of Lie Groups and Related Topics. A.M. Vershik and D.P. Zhelobenko (Eds). Advanced Studies in Contemporary Mathematics 7. Gordon and Breach, New York, 1990. | MR | Zbl
.[50] The problem of harmonic analysis on the infinite-dimensional unitary group. J. Funct. Anal. 205 (2003) 464-524. | MR | Zbl
.[51] Ergodic unitary invariant measures on the space of infinite Hermitian matrices. Amer. Math. Soc. Transl. Ser. 2 175 (1996) 137-175. | MR | Zbl
and .[52] Mackey analysis of infinite classical motion groups. Pacific J. Math. 150 (1991) 139-166. | MR | Zbl
.[53] The cut-off phenomenon for random reflections. Ann. Probab. 24 (1996) 74-96. | MR | Zbl
.[54] The cut-off phenomenon for random reflections. II. Complex and quaternionic cases. Probab. Theory Related Fields 104 (1996) 181-209. | MR | Zbl
.[55] Random rotations: Characters and random walks on SO(N). Ann. Probab. 22 (1994) 398-423. | MR | Zbl
.[56] Bessel convolution on matrix cones. Compos. Math. 143 (2007) 749-779. | MR | Zbl
.[57] The spectrum of the QCD Dirac operator and chiral random matrix theory: The threefold way. Phys. Rev. Lett. 72 (1994) 2531-2533.
.[58] Spiked models in Wishart ensemble. PhD thesis. Available at arXiv:math-pr/0804.0889v1. | MR
.[59] Dyson's Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007) 573-590. | MR | Zbl
.[60] On the statistical distribution of the widths and spacings of nuclear resonance levels. Proc. Cambridge Philos. Soc. 47 (1951) 790-798. | Zbl
.[61] The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A (1928) 32-52. | JFM
.[62] Compact Lie Groups and Their Representations. Transl. of Math. Monographs 40. AMS, Providence, RI, 1973. | MR | Zbl
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