Defosseux, Manon
Orbit measures, random matrix theory and interlaced determinantal processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1 , p. 209-249
Zbl 1216.15024 | MR 2641777 | 1 citation dans Numdam
doi : 10.1214/09-AIHP314
URL stable : http://www.numdam.org/item?id=AIHPB_2010__46_1_209_0

Classification:  15A52,  17B10
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.
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.

Bibliographie

[1] A. Altland and M. Zirnbauer. Nonstandard symmetry classes in mesoscopic normal/superconducting hybrid structures. Phys. Rev. B 55 (1997) 1142-1161.

[2] Y. Baryshnikov. GUEs and queues. Probab. Theory Related Fields 119 (2001) 256-274. MR 1818248 | Zbl 0980.60042

[3] A. Berenstein and A. Zelevinsky. Tensor product multiplicities and convex polytopes in partition space. J. Geom. Phys. 5 (1988) 453-472. MR 1048510 | Zbl 0712.17006

[4] P. Biane. 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 2276889 | Zbl 1117.81082

[5] E. Borel. Sur les principes de la théorie cinétique des gaz. Ann. Sci. École Norm. Sup. 23 (1906) 9-32. Numdam | JFM 37.0944.01 | MR 1509063

[6] A. Borodin and G. Olshanski. Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes. Ann. of Math. (2) 161 (2005) 1319-1422. MR 2180403 | Zbl 1082.43003

[7] A. Borodin, P.L. Ferrari, M. Präehofer and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055-1080. MR 2363389 | Zbl 1136.82028

[8] A. Borodin. Biorthogonal ensembles. Nuclear Phys. B 536 (1999) 704-732. MR 1663328 | Zbl 0948.82018

[9] E. Brezin, S. Hikami and A.I. Larkin. Level statistics inside the vortex of a superconductor and symplectic random-matrix theory in an external source. Phys. Rev. B 60 (1999) 3589-3602.

[10] E. Brezin and S. Hikami. Intersection numbers from the antisymmetric Gaussian matrix model. J. High Energy Phys. (2008) 7 050, 19. MR 2430134

[11] M.F. Bru. Wishart process. J. Theoret. Probab. 4 (1991) 725-751. MR 1132135 | Zbl 0737.60067

[12] J. Cardy. Network models in class C on arbitrary graphs. Comm. Math. Phys. 258 (2005) 87-102. MR 2166841 | Zbl 1079.81075

[13] M. Caselle and U. Magnea. Random matrix theory and symmetric spaces. Phys. Rep. 394 (2004) 41-156. MR 2049671

[14] H. Cohn, M. Larsen and J. Propp. The shape of a typical boxed plane partition. New York J. Math. 4 (1998) 137-165 (electronic). MR 1641839 | Zbl 0908.60083

[15] B. Collins and P. Sniady. Representation of Lie groups and random matrices. Trans. Amer. Math. Soc. 361 (2009) 3269-3287. MR 2485426 | Zbl 1170.22005

[16] M. Defosseux. Orbit measures and interlaced determinantal point processes. C. R. Math. Acad. Sci. Paris Ser. I 346 (2008) 783-788. MR 2427082 | Zbl 1157.60027

[17] P. Diaconis and M. Shahshahani. 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 841092 | Zbl 0586.60012

[18] A.H. Dooley, J. Repka and N.J. Wildberger. Sums of adjoint orbits. Linear Multilinear Algebra 36 (1993) 79-101. MR 1308911 | Zbl 0797.15010

[19] A.K. Duli and N.D. Wildberger. Harmonic analysis and the global exponential map for compact Lie groups. Funct. Anal. Appl. 27 (1993) 25-32. MR 1225907 | Zbl 0804.22011

[20] F.J. Dyson. The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3 (1962) 1199-1215. MR 177643 | Zbl 0134.45703

[21] P. Eichelsbacher and M. Stolz. Large deviations for random matrix ensembles in mesoscopic physics. Available at arXiv:math/0610811v2. MR 2437529 | Zbl 1151.60012

[22] J. Faraut. Infinite Dimensional Spherical Analysis. COE Lecture Note 10. Kyushu University, The 21st Century COE Program “DMHF,” Fukuoka, 2008. MR 2391335 | Zbl 1154.43008

[23] P.J. Forrester. Log-gases and random matrices. To appear. Available at http://www.ms.unimelb.edu.au/~matpjf/matpjf.html. MR 2641363 | Zbl 1217.82003

[24] P.J. Forrester and E. Nordenstam. The anti-symmetric GUE minor process. Available at arXiv:math-pr/0804.3293v1. MR 2663989 | Zbl 1191.15032

[25] M. Fulmek and C. Krattenthaler. Lattice paths proofs for determinantal formulas for symplectic and orthogonal characters. J. Combin. Theory Ser. A 77 (1997) 3-50. MR 1426737 | Zbl 0867.05083

[26] W. Fulton. Young Tableaux. London Mathematical Society Student Text 35. Cambridge Univ. Press, Cambridge, 1997. MR 1464693 | Zbl 0878.14034

[27] I.M. Gelfand and M.L. Tsetlin. Finite dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk. USSR 71 (1981) 275-290.

[28] F. Gillet. Asymptotic behaviour of watermelons. Preprint, 2003. Available at arXiv:math.PR/0307204.

[29] N.R. Goodman. Statistical analysis based on a certain multivariate complex Gaussian distribution (An Introduction). Ann. Math. Statist. 34 (1963) 152-177. MR 145618 | Zbl 0122.36903

[30] G.J. Heckman. Projections of orbits and asymptotic behaviour of multiplicities for compact connected Lie groups. Invent. Math. 67 (1982) 333-356. MR 665160 | Zbl 0497.22006

[31] P. Heinzner, A. Huckleberry and M.R. Zirnbauer. Symmetry classes of disordered fermions. Commun. Math. Phys. 257 (2005) 725-771. MR 2164950 | Zbl 1092.82020

[32] S. Helgason. Groups and Geometric Analysis. Academic Press, New York, 1984. MR 754767 | Zbl 0543.58001

[33] K. Johansson. Random matrices and determinantal processes. Available at arXiv:math-ph/0510038v1. MR 2581882

[34] K. Johansson and E. Nordenstam. Eigenvalues of GUE minors. Electron. J. Probab. 11 (2006) 1342-1371. MR 2268547 | Zbl 1127.60047

[35] M. Kashiwara. On crystal bases. In Representations of Groups. CMS Conference Proceedings 16 155-197. Amer. Math. Soc., Providence, RI, 1995. MR 1357199 | Zbl 0851.17014

[36] M. Kashiwara and T. Nakashima. Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165 (1994) 295-345. MR 1273277 | Zbl 0808.17005

[37] M. Katori, H. Tanemura, T. Nagao and N. Komatsuda. Vicious walk with a wall, noncolliding meanders, chiral and Bogoliubov-de Gennes random matrices. Phys. Rev. E 68 (2003) 1-16.

[38] M. Katori and H. Tanemura. Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45 (2004) 3058-3085. MR 2077500 | Zbl 1071.82045

[39] A.A. Kirillov. Merits and demerits of the orbit method. Bull. Amer. Math. Soc. (N.S.) 36 (1999) 433-488. MR 1701415 | Zbl 0940.22013

[40] A.A. Kirillov. Lectures on the Orbit Method. Graduate Studies in Mathematics 64. Amer. Math. Soc., Providence, RI, 2004. MR 2069175 | Zbl pre02121486

[41] A. Klyachko. Random walks on symmetric spaces and inequalities for matrix spectra. Linear Algebra Appl. 319 (2000) 37-59. MR 1799623 | Zbl 0980.15015

[42] A.W. Knapp. Lie Groups, Beyond an Introduction, 2nd edition. Progress in Mathematics 140. Birkhäuser, Boston, 2002. MR 1920389 | Zbl 1075.22501

[43] C. Krattenthaler, A.J. Guttmann and X.G. Viennot. Vicious walkers, friendly walkers and Young tableaux. II. With a wall. J. Phys. A 33 (2000) 8835-8866. MR 1801472 | Zbl 0970.82016

[44] M.L. Mehta and N. Rosenzweig. Distribution laws for the roots of a random antisymmetric hermitian matrix. Nuclear Phys. A 109 (1968) 449-456.

[45] M.L. Mehta. Random Matrices, 3rd edition. Pure and Applied Mathematics (Amsterdam) 142. Elsevier-Academic Press, Amsterdam, 2004. MR 2129906 | Zbl 1107.15019

[46] T. Nakashima. Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras. Comm. Math. Phys. 154 (1993) 215-243. MR 1224078 | Zbl 0795.17016

[47] A. Okounkov and N. Reshetikhin. The birth of random matrix. Moscow Math. J. 6 (2006) 553-566. MR 2274865 | Zbl 1130.15014

[48] G. Olshanski. Unitary representations of (G, K)-pairs that are connected with the infinite symmetric group S(∞). Leningrad Math. J. 1 (1990) 983-1014. MR 1027466 | Zbl 0731.20009

[49] G. Olshanski. 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 1104279 | Zbl 0724.22020

[50] G. Olshanski. The problem of harmonic analysis on the infinite-dimensional unitary group. J. Funct. Anal. 205 (2003) 464-524. MR 2018416 | Zbl 1036.43002

[51] G. Olshanski and A. Vershik. Ergodic unitary invariant measures on the space of infinite Hermitian matrices. Amer. Math. Soc. Transl. Ser. 2 175 (1996) 137-175. MR 1402920 | Zbl 0853.22016

[52] D. Pickrell. Mackey analysis of infinite classical motion groups. Pacific J. Math. 150 (1991) 139-166. MR 1120717 | Zbl 0739.22016

[53] U. Porod. The cut-off phenomenon for random reflections. Ann. Probab. 24 (1996) 74-96. MR 1387627 | Zbl 0854.60068

[54] U. Porod. The cut-off phenomenon for random reflections. II. Complex and quaternionic cases. Probab. Theory Related Fields 104 (1996) 181-209. MR 1373375 | Zbl 0865.60005

[55] J.S. Rosenthal. Random rotations: Characters and random walks on SO(N). Ann. Probab. 22 (1994) 398-423. MR 1258882 | Zbl 0799.60007

[56] M. Roesler. Bessel convolution on matrix cones. Compos. Math. 143 (2007) 749-779. MR 2330446 | Zbl 1115.33011

[57] J.J.M. Verbaarschot. The spectrum of the QCD Dirac operator and chiral random matrix theory: The threefold way. Phys. Rev. Lett. 72 (1994) 2531-2533.

[58] D. Wang. Spiked models in Wishart ensemble. PhD thesis. Available at arXiv:math-pr/0804.0889v1. MR 2711517

[59] J. Warren. Dyson's Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007) 573-590. MR 2299928 | Zbl 1127.60078

[60] E.P. Wigner. On the statistical distribution of the widths and spacings of nuclear resonance levels. Proc. Cambridge Philos. Soc. 47 (1951) 790-798. Zbl 0044.44203

[61] J. Wishart. The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A (1928) 32-52. JFM 54.0565.02

[62] D.P. Zhelobenko. Compact Lie Groups and Their Representations. Transl. of Math. Monographs 40. AMS, Providence, RI, 1973. MR 473098 | Zbl 0272.22006