Interlaced processes on the circle
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, p. 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 : https://doi.org/10.1214/08-AIHP305
Classification:  60J99,  60B15,  82B21,  05E10
@article{AIHPB_2009__45_4_1165_0,
     author = {Metcalfe, Anthony P. and O'Connell, Neil and Warren, Jon},
     title = {Interlaced processes on the circle},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     pages = {1165-1184},
     doi = {10.1214/08-AIHP305},
     zbl = {1218.60075},
     mrnumber = {2572170},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIHPB_2009__45_4_1165_0/

[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 2483738 | Zbl pre05347023

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

[3] P. H. Berard. Spectres et groupes cristallographiques. I. Domaines euclidiens. Invent. Math. 58 (1980) 179. | MR 570879 | Zbl 0434.35068

[4] Ph. Biane, Ph. Bougerol and N. O'Connell. Littleman paths and Brownian paths. Duke Math. J. 130 (2005) 127-167. | MR 2176549 | Zbl 1161.60330

[5] C. Boutillier. The bead model and limit behaviors of dimer models. Ann. Probab. To appear. | MR 2489161 | Zbl 1171.82006

[6] C. Boutillier and B. De Tilière. Loops statistics in the toroidal honeycomb dimer model. Available at arXiv:math/0608600. | Zbl 1179.60065

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

[8] P. Diaconis. Patterns in eigenvalues: The 70th Josiah Williard Gibbs lecture. Bull. Amer. Math. Soc. 40 (2003) 155-178. | MR 1962294 | Zbl 1161.15302

[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 841092 | Zbl 0586.60012

[10] P. Diaconis and J. A. Fill. Strong stationary times via a new form of duality. Ann. Probab. 18 (1990) 1483-1522. | MR 1071805 | Zbl 0723.60083

[11] F. Dyson. Statistical theory of the energy levels of complex systems, I-III. J. Math. Phys. 3 (1962). | MR 143556

[12] F. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191-1198. | MR 148397 | Zbl 0111.32703

[13] J. Faraut. Analysis on Lie Groups: An Introduction. Cambridge Univ. Press, 2008. | MR 2426516 | Zbl 1147.22001

[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 2219210 | Zbl 1117.15026

[16] W. Fulton. Young Tableaux: With Applications to Representation Theory and Geometry. Cambridge Univ. Press, 1997. | MR 1464693 | Zbl 0878.14034

[17] I. Gessel and R. Zeilberger. Random walk in a Weyl chamber. Proc. Amer. Math. Soc. 115 (1992) 27-31. | MR 1092920 | Zbl 0792.05148

[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 1830441 | Zbl 0989.82030

[19] J. Gunson. Proof of a conjecture of Dyson in the statistical theory of energy levels. J. Math. Phys. 3 (1962) 752-753. | MR 148401 | Zbl 0111.43903

[20] D. Hobson and W. Werner. Non-colliding Brownian motions on the circle. Bull. London Math. Soc. 28 (1996) 643-650. | MR 1405497 | Zbl 0853.60060

[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 2268547 | Zbl 1127.60047

[23] R. Kenyon, A. Okounkov and S. Sheffield. Dimers and amoebae. Ann. Math. 163 (2006) 1019-1056. | MR 2215138 | Zbl 1154.82007

[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 1887625 | Zbl 1007.60075

[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 2349573 | Zbl 1139.60017

[26] P. Mcnamara. Cylindric skew Schur functions. Adv. Math. 205 (2006) 275-312. | MR 2254313 | Zbl 1110.05099

[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 1990168 | Zbl 1031.05132

[29] N. O'Connell. Conditioned random walks and the RSK correspondence. J. Phys. A 36 (2003) 3049-3066. | MR 1986407 | Zbl 1035.05097

[30] N. O'Connell and M. Yor. A representation for non-colliding random walks. Electron. Comm. Probab. 7 (2002) 1-12. | MR 1887169 | Zbl 1037.15019

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

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

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

[34] A. Postnikov. Affine approach to quantum Schubert calculus. Duke Math. J. 128 (2005) 473-509. | MR 2145741 | Zbl 1081.14070

[35] L. C. G. Rogers and J. Pitman. Markov functions. Ann. Probab. 9 (1981) 573-582. | MR 624684 | Zbl 0466.60070

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

[37] F. Toomey. Bursty traffic and finite capacity queues. Ann. Oper. Res. 79 (1998) 45-62. | MR 1630874 | Zbl 0896.90099

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

[39] R. J. Williams. Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 (1987) 459-485. | MR 894900 | Zbl 0608.60074