Interlaced processes on the circle

Anthony P. Metcalfe; Neil O’Connell; Jon Warren

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 4, page 1165-1184
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Metcalfe, Anthony P., O’Connell, Neil, and Warren, Jon. "Interlaced processes on the circle." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 1165-1184. <http://eudml.org/doc/78059>.

@article{Metcalfe2009,
abstract = {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.},
author = {Metcalfe, Anthony P., O’Connell, Neil, Warren, Jon},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {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; Pitman's theorem; reflected Brownian motion; Brownian motion in an alcove},
language = {eng},
number = {4},
pages = {1165-1184},
publisher = {Gauthier-Villars},
title = {Interlaced processes on the circle},
url = {http://eudml.org/doc/78059},
volume = {45},
year = {2009},
}

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
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 1165
EP - 1184
AB - 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.
LA - eng
KW - 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; Pitman's theorem; reflected Brownian motion; Brownian motion in an alcove
UR - http://eudml.org/doc/78059
ER -

References

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  1. [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. Zbl1215.82003MR2483738
  2. [2] Y. Baryshnikov. GUEs and queues. Probab. Theory Related Fields 119 (2001) 256–274. Zbl0980.60042MR1818248
  3. [3] P. H. Berard. Spectres et groupes cristallographiques. I. Domaines euclidiens. Invent. Math. 58 (1980) 179. Zbl0434.35068MR570879
  4. [4] Ph. Biane, Ph. Bougerol and N. O’Connell. Littleman paths and Brownian paths. Duke Math. J. 130 (2005) 127–167. Zbl1161.60330MR2176549
  5. [5] C. Boutillier. The bead model and limit behaviors of dimer models. Ann. Probab. To appear. Zbl1171.82006MR2489161
  6. [6] C. Boutillier and B. de Tilière. Loops statistics in the toroidal honeycomb dimer model. Available at arXiv:math/0608600. Zbl1179.60065
  7. [7] M. Defosseux. Orbit measures and interlaced determinantal point processes. C. R. Math. Acad. Sci. Paris 346 (2008) 783–788. Zbl1157.60027MR2427082
  8. [8] P. Diaconis. Patterns in eigenvalues: The 70th Josiah Williard Gibbs lecture. Bull. Amer. Math. Soc. 40 (2003) 155–178. Zbl1161.15302MR1962294
  9. [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. Zbl0586.60012MR841092
  10. [10] P. Diaconis and J. A. Fill. Strong stationary times via a new form of duality. Ann. Probab. 18 (1990) 1483–1522. Zbl0723.60083MR1071805
  11. [11] F. Dyson. Statistical theory of the energy levels of complex systems, I–III. J. Math. Phys. 3 (1962). Zbl0105.41604MR143556
  12. [12] F. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191–1198. Zbl0111.32703MR148397
  13. [13] J. Faraut. Analysis on Lie Groups: An Introduction. Cambridge Univ. Press, 2008. Zbl1147.22001MR2426516
  14. [14] P. J. Forrester and T. Nagao. Determinantal correlations for classical projection processes. Available at arXiv:0801.0100. Zbl0917.15018
  15. [15] P. Forrester and E. Rains. Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices. Int. Math. Res. Not. (2006) 48306. Zbl1117.15026MR2219210
  16. [16] W. Fulton. Young Tableaux: With Applications to Representation Theory and Geometry. Cambridge Univ. Press, 1997. Zbl0878.14034MR1464693
  17. [17] I. Gessel and R. Zeilberger. Random walk in a Weyl chamber. Proc. Amer. Math. Soc. 115 (1992) 27–31. Zbl0792.05148MR1092920
  18. [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. Zbl0989.82030MR1830441
  19. [19] J. Gunson. Proof of a conjecture of Dyson in the statistical theory of energy levels. J. Math. Phys. 3 (1962) 752–753. Zbl0111.43903MR148401
  20. [20] D. Hobson and W. Werner. Non-colliding Brownian motions on the circle. Bull. London Math. Soc. 28 (1996) 643–650. Zbl0853.60060MR1405497
  21. [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. [22] K. Johansson and E. Noordenstam. Eigenvalues of GUE minors. Elect. J. Probab. 11 (2006) 1342–1371. Zbl1127.60047MR2268547
  23. [23] R. Kenyon, A. Okounkov and S. Sheffield. Dimers and amoebae. Ann. Math. 163 (2006) 1019–1056. Zbl1154.82007MR2215138
  24. [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). Zbl1007.60075MR1887625
  25. [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. Zbl1139.60017MR2349573
  26. [26] P. McNamara. Cylindric skew Schur functions. Adv. Math. 205 (2006) 275–312. Zbl1110.05099MR2254313
  27. [27] A. P. Metcalfe. Ph.d. thesis. To appear. 
  28. [28] N. O’Connell. A path-transformation for random walks and the Robinson–Schensted correspondence. Trans. Amer. Math. Soc. 355 (2003) 3669–3697. Zbl1031.05132MR1990168
  29. [29] N. O’Connell. Conditioned random walks and the RSK correspondence. J. Phys. A 36 (2003) 3049–3066. Zbl1035.05097MR1986407
  30. [30] N. O’Connell and M. Yor. A representation for non-colliding random walks. Electron. Comm. Probab. 7 (2002) 1–12. Zbl1037.15019MR1887169
  31. [31] A. Okounkov and N. Reshetikhin. The birth of random matrix. Moscow Math. J. 6 (2006) 553–566. Zbl1130.15014MR2274865
  32. [32] U. Porod. The cut-off phenomenon for random reflections Ann. Probab. 24 (1996) 74–96. Zbl0854.60068MR1387627
  33. [33] U. Porod. The out-off phenomenon for random reflections II: Complex and quaternionic cases. Probab. Theory Related Fields 104 (1996) 181–209. Zbl0865.60005MR1373375
  34. [34] A. Postnikov. Affine approach to quantum Schubert calculus. Duke Math. J. 128 (2005) 473–509. Zbl1081.14070MR2145741
  35. [35] L. C. G. Rogers and J. Pitman. Markov functions. Ann. Probab. 9 (1981) 573–582. Zbl0466.60070MR624684
  36. [36] J. S. Rosenthal. Random rotations: Characters and random walks on SO(N). Ann. Probab. 22 (1994) 398–423. Zbl0799.60007MR1258882
  37. [37] F. Toomey. Bursty traffic and finite capacity queues. Ann. Oper. Res. 79 (1998) 45–62. Zbl0896.90099MR1630874
  38. [38] J. Warren. Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007) 573–590. Zbl1127.60078MR2299928
  39. [39] R. J. Williams. Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 (1987) 459–485. Zbl0608.60074MR894900

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