A path-transformation for random walks and the Robinson-Schensted correspondence

Transactions of the American Mathematical Society

Volumn 355, Issue 9, 2003, Pages 3669-3697

  O'Connell, Neil ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Brownian motion; Hermitian Brownian motion; Intertwining; Markov functions; Pitman's representation theorem; Random matrices; Random walk; Robinson Schensted correspondence; RSK; Weyl chamber; Young tableau

Indexed keywords

EID: 0042854799     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-03-03226-4     Document Type: Article

References (44)

1. F. Baccelli, G. Cohen, G.J. Olsder and J.-P. Quadrat. Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, 1992. MR 94b:93001
2. F. Baccelli and W.A. Massey. A sample path analysis of the M/M/1 queue. J. Appl. Probab. 26 (1989) 418-422. MR 90i:60065
3. F. Baccelli and W.A. Massey. A transient analysis of the two-node series Jackson network. INRIA Rapport de Recherche No. 852 (1988).
4. J. Baik, P. Deift arid K. Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), no. 4, 1119-1178. MR 2000e:05006
5. Yu. Baryshnikov. GUES and queues. Probab. Theory Related Fields 119 (2001) 256-274. MR 2002a:60165
6. A. Berenstein and A.N. Kirillov. The Robinson-Schensted-Knuth bijection, quantum matrices and piecewise linear combinatorics. To appear in: Proceedings of the 13th International Conference on Formal Power Series and Algebraic Combinatorics, Arizona, 2001.
7. Ph. Biane. Quelques propriétés du mouvement brownien dans un cone. Stoch. Proc. Appl. 53 (1994), no. 2, 233-240. MR 95j:60129
8. Ph. Biane. Théorème de Ney-Spitzer sur le dual de SU(2). Trans. Amer. Math. Soc. 345, no. 1 (1994) 179-194. MR 95a:60103
9. Ph. Biane. Intertwining of Markov semi-groups, some examples. Séminaire de Probabilités, XXIX, 30-36, Lecture Notes in Math., 1613, Springer-Verlag, Berlin, 1995. MR 98k:46117
10. Ph. Biane. Quantum random walk on the dual of SU(n). Probab. Theory Related Fields 89 (1991), no. 1, 117-129. MR 93a:46119
11. Ph. Bougerol and Th. Jeulin. Paths in Weyl chambers and random matrices. Probab. Theory Related Fields 124 (2002) 517-543.
12. K. Burdzy. A three-dimensional Brownian path reflected on a Brownian path is a free Brownian path. Letters Math. Phys. 27 (1993) 239-241. MR 94c:60132
13. K. Burdzy and D. Nualart. Brownian motion reflected on Brownian motion. Probab. Theory Related Fields, 122 (2002), 471-493.
14. q(gl(n, C)) at q = 0 and the Robinson-Schensted correspondence. Physics and Mathematics of Strings, 185-211, World Sci. Publishing, Teaneck, NJ, 1990. MR 92h:17012
15. J.L. Doob. Classical Potential Theory and its Probabilistic Counterpart. Springer-Verlag, 1984. MR 85k:31001
16. Y. Doumerc. An asymptotic link between LUE and GUE and its spectral interpretation. Preprint.
17. F. J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191-1198. MR 26:5904
18. William Fulton. Young Tableaux. London Mathematical Society student texts: 35. Cambridge University Press, 1997. MR 99f:05119
19. P.W. Glynn and W. Whitt. Departures from many queues in series. Ann. Appl. Prob. l (1991), no. 4, 546-572. MR 921:60162
20. D. Grabiner. Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. IHP Probab. Statist. 35 (1999), no. 2, 177-204. MR 20001:60091
21. J. Gravner, C.A. Tracy and H. Widom. Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102 (2001), 1085-1132. MR 2002d:82065
22. Harish-Chandra. Invariant differential operators on a semisimple Lie algebra. Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 252-253. MR 18:218c
23. J. M. Harrison and R. J. Williams. On the quasireversibility of a multiclass Brownian service station. Ann. Probab. 18 (1990) 1249-1268. MR 91i:60204
24. A.R. Its, C.A. Tracy and H. Widom. Random words, Toeplitz determinants, and integrable systems. In: Random Matrices and their Applications, MSRI Publications, Volume 40, 2001, 245-258. MR 20021:82040
25. C. Itzykson and J.B. Zuber. The planar approximation. II. J. Math. Phys. 21 (1980), no. 3, 411-421. MR 81a:81068
26. K. Johansson. Shape fluctuations and random matrices. Commun. Math. Phys. 209 (2000) 437-476. MR 2001h:60177
27. K. Johansson. Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. (2) 153 (2001), no. 1, 259-296. MR 2002g:05188
28. Wolfgang König, Neil O'Connell and Sebastien Roch. Non-colliding random walks, tandem queues and the discrete ensembles. Elect. J. Probab. 7 (2002) Paper no. 5, 1-24.
29. Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon. The plactic monoid. In: Algebraic Combinatorics on Words, Cambridge University Press, 2002.
30. Peter Littelmann. The path model, the quantum Frobenius map and standard monomial theory. Algebraic groups and their representations (Cambridge, 1997), 175-212, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 517, Kluwer Acad. Publ., Dordrecht, 1998. MR 99m:20096
31. I.G. MacDonald. Symmetric Functions and Hall Polynomials. Second edition, Oxford, 1995. MR 96h:05207
32. M.L. Mehta. Random Matrices: Second Edition. Academic Press, 1991. MR 92f:82002
33. Neil O'Connell. Random matrices, non-colliding processes and queues. Lecture Notes in Math., vol. 1801, Springer-Verlag, 2003.
34. Neil O'Connell. Conditioned random walks and the RSK correspondence. J. Phys. A 36 (2003) 3049-3066.
35. Neil O'Connell and Marc Yor. Brownian analogues of Burke's theorem. Stoch. Proc. Appl. 96 (2) (2001) pp. 285-304. MR 2002h:60175
36. Neil O'Connell and Marc Yor. A representation for non-colliding random walks. Elect. Commun. Probab. 7 (2002) 1-12.
37. J.W. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab. 7 (1975) 511-526. MR 51:11677
38. L.C.G. Rogers and J.W. Pitman. Markov functions. Ann. Probab. 9 (1981) 573-582. MR 82j:60133
39. F. Soucaliuc, B. Toth and W. Werner. Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. Henri Poincaré 36 (2000) 509-545. MR 2002a:60139
40. R.P. Stanley. Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999. MR 2000k:05026
41. C.A. Tracy and H. Widom. Fredholm determinants, differential equations and matrix models. Comm. Math. Phys. 163 (1994), no. 1, 33-72. MR 95e:82005
42. C.A. Tracy and H. Widom. On the distributions of the lengths of the longest monotone subsequences in random words. Probab. Theory Related Fields 119 (2001), no. 3, 350-380. MR 2002a:60013
43. David Williams. Diffusions, Markov Processes and Martingales. Volume 1: Foundations. Wiley, 1979. MR 80i:60100
44. David Williams. Path decomposition and continuity of local time for one-dimensional diffusions I. Proc. London Math. Soc. 28 (1974), no. 3, 738-768. MR 50:3373