Interlaced processes on the circle
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1165-1184.

Quand deux opérateurs de Markov commutent, cela suggère que nous pouvons coupler deux copies d'un des processus correspondants. Nous construisons explicitement un certain nombre de couplages de ce type pour une famille de processus de Markov qui commutent sur l'ensemble des classes de conjugaison du groupe unitaire. Nous utilisons, à cette fin, une règle dynamique inspirée par l'algorithme RSK. Notre motivation est de développer un programme parallèle sur le cercle, pour des connections récemment mises à jour dans la théorie des matrices aléatoires entre des systèmes de particules réfléchies et conditionnées sur la droite. Une des chaînes de Markov que nous considérons donne lieu à une famille de mesures de Gibbs sur des configurations de perles sur le cylindre infini. Nous prouvons que ces mesures ont la structure déterminantale et calculons le noyau de corrélation espace-temps correspondant.

When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on “bead configurations” on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel.

DOI : 10.1214/08-AIHP305
Classification : 60J99, 60B15, 82B21, 05E10
Mots-clés : random matrices, RSK correspondence, coupling, interlacing, rank 1 perturbation, random reflection, Pitman's theorem, reflected brownian motion, brownian motion in an alcove, bead model on a cylinder, determinantal point process, random tiling, dimer configuration 
@article{AIHPB_2009__45_4_1165_0,
     author = {Metcalfe, Anthony P. and O{\textquoteright}Connell, Neil and Warren, Jon},
     title = {Interlaced processes on the circle},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1165--1184},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     doi = {10.1214/08-AIHP305},
     mrnumber = {2572170},
     zbl = {1218.60075},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP305/}
}
TY  - JOUR
AU  - Metcalfe, Anthony P.
AU  - O’Connell, Neil
AU  - Warren, Jon
TI  - Interlaced processes on the circle
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 1165
EP  - 1184
VL  - 45
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP305/
DO  - 10.1214/08-AIHP305
LA  - en
ID  - AIHPB_2009__45_4_1165_0
ER  - 
%0 Journal Article
%A Metcalfe, Anthony P.
%A O’Connell, Neil
%A Warren, Jon
%T Interlaced processes on the circle
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 1165-1184
%V 45
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP305/
%R 10.1214/08-AIHP305
%G en
%F AIHPB_2009__45_4_1165_0
Metcalfe, Anthony P.; O’Connell, Neil; Warren, Jon. Interlaced processes on the circle. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1165-1184. doi : 10.1214/08-AIHP305. http://www.numdam.org/articles/10.1214/08-AIHP305/

[1] D. Bakry and N. Huet. The hypergroup property and representation of Markov kernels. In Séminaire de Probabilités XLI. Lecture Notes in Mathematics 1934. Springer, 2008. | MR

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

[3] P. H. Berard. Spectres et groupes cristallographiques. I. Domaines euclidiens. Invent. Math. 58 (1980) 179. | MR | Zbl

[4] Ph. Biane, Ph. Bougerol and N. O'Connell. Littleman paths and Brownian paths. Duke Math. J. 130 (2005) 127-167. | MR | Zbl

[5] C. Boutillier. The bead model and limit behaviors of dimer models. Ann. Probab. To appear. | MR | Zbl

[6] C. Boutillier and B. De Tilière. Loops statistics in the toroidal honeycomb dimer model. Available at arXiv:math/0608600. | Zbl

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

[8] P. Diaconis. Patterns in eigenvalues: The 70th Josiah Williard Gibbs lecture. Bull. Amer. Math. Soc. 40 (2003) 155-178. | MR | Zbl

[9] P. Diaconis and M. Shahshahani. Products of random matrices as they arise in the study of random walks on groups. Contemp. Math. 50 (1986) 183-195. | MR | Zbl

[10] P. Diaconis and J. A. Fill. Strong stationary times via a new form of duality. Ann. Probab. 18 (1990) 1483-1522. | MR | Zbl

[11] F. Dyson. Statistical theory of the energy levels of complex systems, I-III. J. Math. Phys. 3 (1962). | MR

[12] F. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191-1198. | MR | Zbl

[13] J. Faraut. Analysis on Lie Groups: An Introduction. Cambridge Univ. Press, 2008. | MR | Zbl

[14] P. J. Forrester and T. Nagao. Determinantal correlations for classical projection processes. Available at arXiv:0801.0100.

[15] P. Forrester and E. Rains. Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices. Int. Math. Res. Not. (2006) 48306. | MR | Zbl

[16] W. Fulton. Young Tableaux: With Applications to Representation Theory and Geometry. Cambridge Univ. Press, 1997. | MR | Zbl

[17] I. Gessel and R. Zeilberger. Random walk in a Weyl chamber. Proc. Amer. Math. Soc. 115 (1992) 27-31. | MR | Zbl

[18] J. Gravner, C. Tracy and H. Widom. Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102 (2001) 1085-1132. | MR | Zbl

[19] J. Gunson. Proof of a conjecture of Dyson in the statistical theory of energy levels. J. Math. Phys. 3 (1962) 752-753. | MR | Zbl

[20] D. Hobson and W. Werner. Non-colliding Brownian motions on the circle. Bull. London Math. Soc. 28 (1996) 643-650. | MR | Zbl

[21] K. Johansson. Random matrices and determinantal processes. Lectures given at the summer school on Mathematical statistical mechanics in July 05 at Ecole de Physique, Les Houches. Available at arXiv:math-ph/0510038.

[22] K. Johansson and E. Noordenstam. Eigenvalues of GUE minors. Elect. J. Probab. 11 (2006) 1342-1371. | MR | Zbl

[23] R. Kenyon, A. Okounkov and S. Sheffield. Dimers and amoebae. Ann. Math. 163 (2006) 1019-1056. | MR | Zbl

[24] W. König, N. O'Connell and S. Roch. Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7 (2002). | MR | Zbl

[25] L. Kruk, J. Lehoczky, K. Ramanan and S. Shreve. An explicit formula for the Skorohod map on [0, a]. Ann. Probab. 35 (2007) 1740-1768. | MR | Zbl

[26] P. Mcnamara. Cylindric skew Schur functions. Adv. Math. 205 (2006) 275-312. | MR | Zbl

[27] A. P. Metcalfe. Ph.d. thesis. To appear.

[28] N. O'Connell. A path-transformation for random walks and the Robinson-Schensted correspondence. Trans. Amer. Math. Soc. 355 (2003) 3669-3697. | MR | Zbl

[29] N. O'Connell. Conditioned random walks and the RSK correspondence. J. Phys. A 36 (2003) 3049-3066. | MR | Zbl

[30] N. O'Connell and M. Yor. A representation for non-colliding random walks. Electron. Comm. Probab. 7 (2002) 1-12. | MR | Zbl

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

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

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

[34] A. Postnikov. Affine approach to quantum Schubert calculus. Duke Math. J. 128 (2005) 473-509. | MR | Zbl

[35] L. C. G. Rogers and J. Pitman. Markov functions. Ann. Probab. 9 (1981) 573-582. | MR | Zbl

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

[37] F. Toomey. Bursty traffic and finite capacity queues. Ann. Oper. Res. 79 (1998) 45-62. | MR | Zbl

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

[39] R. J. Williams. Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 (1987) 459-485. | MR | Zbl

Cité par Sources :