Random matrix central limit theorems for nonintersecting random walks

Annals of Probability

Volumn 35, Issue 5, 2007, Pages 1807-1834

  Baik, Jinho ^a   Suidan, Toufic M ^b  

^a UNIVERSITY OF MICHIGAN   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Nonintersecting random; Riemann Hilbert problem; Sine kernel; Stieltjes Wigert polynomials; Strong approximation; Tracy Widom distribution; Walks

Indexed keywords

EID: 49949101360     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117906000001105     Document Type: Article

References (55)

1. ADLER, M. and VAN MOERBEKE P. (2005). PDEs for the joint distributions of the Dyson, Airy and sine processes. Ann. Probab. 33 1326-1361. MR2150191
2. ALDOUS, D. and DIACONIS, P. (1999). Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem. Bull. Amer. Math. Soc. (N.S.) 36 413-432. MR1694204
3. BAIK, J. (2000). Random vicious walks and random matrices. Comm. Pure Appl. Math. 53 1185-1410. MR1773413
4. BAIK, J., BORODIN, A., DEIFT, P. and SUIDAN, T. (2006). A model for the bus system in Cuernevaca (Mexico). J. Phys. A 39 8965-8975.
5. BAIK, J., DEIFT, P. and JOHANSSON, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119-1178. MR1682248
6. BAIK, J., DEIFT, P. and JOHANSSON, K. (2000). On the distribution of the length of the second row of a Young diagram under Plancherel measure. Geom. Funct. Anal. 10 702-731. MR1791137
7. BAIK, J., DEIFT, P. and RAINS, E. M. (2001). A Fredholm determinant identity and the convergence of moments for random Young tableaux. Comm. Math. Phys. 223 627-672. MR1866169
8. BAIK, J., KRIECHERBAUER, T., MCL AUGHLIN, K. and MILLER, P. (2003). Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: Announcement of results. Int. Math. Res. Not. 821-858. MR1952523
9. BAIK, J., KRIECHERBAUER, T., MCL AUGHLIN, K. and MILLER, P. (2007). Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles. Princeton Univ. Press.
10. BAIK, J. and SUIDAN, T. (2005). A GUE central limit theorem and universality of directed last and first passage site percolation. Int. Math. Res. Not. 325-337. MR2131383
11. BLEHER, P. and ITS, A. (1999). Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 185-266. MR1715324
12. BODINEAU, T. and MARTIN, J. (2005). A universality property for last-passage percolation paths close to the axis. Electron. Comm. Probab. 10 105-112. MR2150699
13. BORODIN, A., OKOUNKOV, A. and OLSHANSKI, G. (2000). On asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 481-515. MR1758751
14. BORODIN, A. and RAINS, E. (2005). Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 291-317. MR2185331
15. DEIFT, P. (1999). Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. CIMS, New York.
16. DEIFT, P. (2000). Integrable systems and combinatorial theory. Notices Amer. Math. Soc. 47 631-640. MR1764262
17. DEIFT, P. and GIOEV, D. (2004). Universality in random matrix theory for orthogonal and symplectic ensembles. Available at http://lanl.arxiv.org/abs/math-ph/0411.075.
18. DEIFT, P. and. GIOEV, D. (2007). Universality at the edge of the spectrum, for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. To appear. Available at http://lanl.arxiv.org/abs/math-ph/0507023.
19. DEIFT, P., KRIECHERBAUER, T. and MCLAUGHLIN, K. (1998). New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95 388-475. MR1657691
20. DEIFT, P., KRIECHERBAUER, T., MCLAUGHLIN, K., VENAKIDES, S. and ZHOU, X. (1999). Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 1491-1552. MR1711036
21. DEIFT, P., KRIECHERBAUER, T., MCLAUGHLIN, K., VENAKIDES, S. and ZHOU, X. (1999). Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 1335-1425. MR1702716
22. DYSON, F. (1962). A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 1191-1198. MR0148397
23. EYNARD, B. and MEHTA, M. L. (1998). Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 314449-4456. MR1628667
24. FISHER, M. (1984). Walks, walls, wetting, and melting. J. Statist. Phys. 34 667-729. MR0751710
25. FOKAS, A., ITS, A. and KITAEV, V. (1991). Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 313-344. MR1137067
26. FORRESTER, P. (1989). Vicious random walks in the limit of large number of walks. J. Statist. Phys. 56 767-782. MR1015891
27. FORRESTER, P. (2001). Random walks and random permutations. J. Phys. A 34 L417-L423. MR1862639
28. FORRESTER, P. (1994). Properties of an exact crystalline many-body ground state. J. Statist. Phys. 76 331-346. MR1297879
29. ISMAIL, M. (2005). Asymptotics of q-orthogonal polynomials and a q-Airy function. Int. Math. Res. Not. 1063-1088. MR2149641
30. JOHANSSON, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437-476. MR1737991
31. JOHANSSON, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 259-296. MR1826414
32. JOHANSSON, K. (2001). Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 683-705. MR1810949
33. JOHANSSON, K. (2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 225-280. MR1900323
34. JOHANSSON, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277-329. MR2018275
35. KARLIN, S. (1988). Coincident probabilities and applications in combinatorics. J. Appl. Probab. 25A 185-200. MR0974581
36. KARLIN, S. and MCGREGOR, J. (1959). Coincidence probability. Pacific J. Math. 9 1141-1164. MR0114248
37. KOEKOEK, R. and SWARTTOUW, R. (1998). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Available at http://aw.twi.tudel.ft.nl/~koekoek/askey.html.
38. KOMLÓS, J., MAJOR, P. and TUSNÁ DY, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111-131. MR0375412
39. KOMLÓS, J., MAJOR, P. and TUSNÁ DY, G. (1976). An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33-58. MR0402883
40. KUIJLAARS, A. and VANLESSEN, M. (2003). Universality for eigenvalue correlations at the origin of the spectrum. Comm. Math. Phys. 243 163-191. MR2020225
41. MEHTA, M. (1991). Random Matrices, 2nd ed. Academic Press, Boston, MA. MR1083764
42. O'CONNELL, N. and YOR, M. (2002). A representation for non-colliding random walks. Electron. Comm. Probab. 7 1-12. MR1887169
43. OKOUNKOV, A. (2000). Random matrices and random permutations. Int. Math. Res. Not. 1043-1095. MR1802530
44. PASTUR, L. and SHCHERBINA, M. (1997). Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Statist. Phys. 86 109-147. MR1435193
45. PRÄHOFER, M. and SPOHN, H. (2000). Universal distributions for growth processes in 1 + 1 dimensions and random matrices. Phys. Rev. Lett. 84 4882-4885.
46. PRÄHOFER, M. and. SPOHN, H. (2002). Scale invariance of the PNG Droplet and the Airy process. J. Statist. Phys. 108 1071-1106. MR1933446
47. SAFF, E. B. and TOTIK, V. (1997). Logarithmic Potentials with External Fields. Springer, New York. MR1485778
48. SOSHNIKOV, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697-733. MR1727234
49. STANLEY, R. P. (1999). Enumerative Combinatorics 2. Cambridge Univ. Press. MR1676282
50. SUIDAN, T. (2006). A Remark on Chatterjee's theorem and last passage percolation. J. Phys. A 39 8977-8981. MR2240468
51. SZEGÖ, G. (1975). Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI. MR0372517
52. TRACY, C. and WIDOM, H. (1998). Correlation functions, cluster functions and spacing distributions for random matrices. J. Statist. Phys. 92 809-835. MR1657844
53. TRACY, C. and WIDOM, H. (2003). A system of differential equations for the Airy process. Electron. Comm. Probab. 8 93-98. MR1987098
54. VANLESSEN, M. (2007). Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random, matrix theory. Constr. Approx. To appear. Available at http://lanl.arxiv.org/abs/math.CA/0504604.
55. WANG, Z. and WONG, R. (2004). Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 698-718. MR2229666