Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture

Bulletin of the American Mathematical Society

Volumn 40, Issue 2, 2003, Pages 155-178

  Diaconis, Persi ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037396045     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-03-00975-3     Document Type: Article

References (97)

1. Adler, M.; Van Moerbeke, P., Integrals over Classical Groups, Random Permutations, Toda and Toeplitz Lattices. Comm. Pure Appl. Math. 2001, 54, 153-205.
2. Adler, M.; Van Moerbeke, P., Recursion Relations for Unitary Integrals, Combinatorics and the Toeplitz Lattice. Technical Report, Dept. of Mathematics, Brandeis, 2002.
3. Adler, M.; Shiota, T.; Van Moerbeke, P., Random Matrices, Virasoro Algebras, and Non-Commutative KP. Duke Math. J. 1998, 94, 379-431. MR 99e:58088
4. Aldous, D.; Diaconis, P., Longest Increasing Subsequences: From Patience Sorting to the Baik-Deift-Johansson Theorem. Bull. Amer. Math. Soc. 1999, 36, 413-432. MR 2000g:60013
5. Anderson, T., Asymptotic Theory for Principal Component Analysis. Ann. Math. Statist. 1963, 34, 122-148. MR 26:3149
6. d'Aristotle, A., An Invariance Principle for Triangular Arrays. Jour. Theoret. Probab. 2000, 13, 327-342.
7. d'Aristotle, A.; Diaconis, P; Newman, C., Brownian Motion and the Classical Groups. Technical Report, Stanford University, 2002.
8. Bai, Z., Methodologies in Spectral Analysis of Large Dimensional Random Matrices: A Review. Statistica Sinica 1999, 9, 611-677. MR 2000e:60044
9. Baik, J.; Deift, P.; Johansson, K., On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations. Jour. Amer. Math. Soc. 1999, 12, 1119-1178. MR 2000e:05006
10. Basor, E., Distribution Functions for Random Variables for Ensembles of Positive Hermitian Matrices. Comm. Math. Phys. 1997, 188, 327-350. MR 99b:82046
11. Basor, E., Connections Between Random Matrices and Szegö Limit Theorem. In Spectral Problems in Geometry and Arithmetic; Branson, T., Ed.; Amer. Math. Soc.: Providence, RI, 1999; 1-7. MR 2000e:47049
12. Berry, M.; Keating, J., The Riemann Zeros and Eigenvalue Asymptotics. Siam Review 1999, 41, 236-266. MR 2000f:11107
13. Biane, P., Free Probability for Probabilists, 2000, preprint.
14. Bohigas, O.; Giannoni, M., Chaotic Motion and Random Matrix Theories. In Mathematical and Computational Methods in Nuclear Physics; Dehesa, J.L., Ed.; Springer Lecture Notes in Physics, 1984, 209, 1-99. MR 86c:58129
15. Borel, E., Sur les Principes de la Theorie Cinetique des Gaz. Annales, L'Ecole Normal Sup. 1906, 23, 9-32.
16. Borodin, A.; Olshansky, G., Correlation Kernels Arising from the Infinite-Dimensional Unitary Group and Its Representations. University of Pennsylvania, Department of Mathematics, 2001, preprint.
17. Böttcher, A. and Silbermann, B., Introduction to Large Truncated Toeplitz Matrices. Springer-Verlag: New York, 1999. MR 2001b:47043
18. Bougerol, Ph. and Jeulin, Th., Paths in Weyl Chambers and Random Matrices. Laboratoire de Probabilities: Paris 2001, preprint.
19. Boutet de Monvel, A.; Pastur, L.; Shcherbina, M., On the Statistical Mechanics Approach in the Random Matrix Theory: Integrated Density of States. Jour. Statist. Phys. 1995, 79, 585-611. MR 96d:82033
20. Bump, D. and Diaconis, P., Toeplitz Minors. Jour. Combin. Th. A. 2002, 97, 252-271. MR 2002j:47052
21. Bump, D.; Diaconis, P.; Keller, J., Unitary Correlations and the Fejer Kernel. Mathematical Phys., Analysis, Geometry 2002, 5, 101-123.
22. Conrey, B., L-Functions and Random Matrices. In Mathematics Unlimited 2001 and Beyond; Enquist, B., Schmid, W. Eds.; Springer-Verlag: Berlin, 2001; 331-352.
23. Conrey, B.; Farmer, D.; Keating, J.; Rubinstein, M.; Snaith, W., Correlation of Random Matrix Polynomials. Technical Report, American Institute of Mathematics, 2002.
24. Coram, M.; Diaconis, P., New Tests of the Correspondence Between Unitary Eigenvalues and the Zeros of Riemann's Zeta Function. Jour. Phys. A. 2002, to appear.
25. Daley, D.; Verre-Jones, D., An Introduction to the Theory of Point Processes. Springer-Verlag: New York, 1988. MR 90e:60060
26. Deift, P., Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes #3, NYU/Courant Institute: New York, and Amer. Math. Soc.: Providence, RI, 1999. MR 2000g:47048
27. Deift, P., Integrable Systems and Combinatorial Theory. Notices, Amer. Math. Soc. 2000, 47, 631-640. MR 2001g:05012
28. Diaconis, P., Group Representations in Probability and Statistics. Ins. Math. Statist., Hayward, CA, 1986. MR 90a:60001
29. Diaconis, P., Applications of the Method of Moments in Probability and Statistics. In Moments in Mathematics; Landau, H., Ed.; Amer. Math. Soc.: Providence, RI, 1987; 125-142. MR 89m:60006
30. Diaconis, P.; Shahshahani, M., Products of Random Matrices as They Arise in the Study of Random Walks on Groups. Contemp. Math. 1986, 50, 183-195. MR 87k:60025
31. Diaconis, P.; Shahshahani, M., The Subgroup Algorithm for Generating Uniform Random Variables. Prob. Eng. and Info. Sci. 1987, 1, 15-32.
32. Diaconis, P.; Shahshahani, M., On the Eigenvalues of Random Matrices. In Studies in Applied Probability; Gani, J., Ed.; Jour. Appl. Probab.: Special Vol. 31A, 1994; 49-62. MR 95m:60011
33. Diaconis, P.; Evans, S., Linear Functional of Eigenvalues of Random Matrices. Transactions Amer. Math. Soc. 2001, 353, 2615-2633. MR 2002d:60003
34. Diaconis, P.; Evans, S., Immanants and Finite Point Processes. Jour. Combin. Th. A. 2000, 91, 305-321. MR 2001m:15018
35. Diaconis, P.; Evans, S., A Different Construction of Gaussian Fields from Markov Chains: Dirichlet Covariances. Ann. Inst. Henri Poincaré, 2002, to appear.
36. Diaconis, P.; Freedman, D., A Dozen deFinetti-Style Results in Search of a Theory. Ann. Inst. Henri Poincaré, 1987, 23, 397-423. MR 88f:60072
37. Diaconis, P.; Eaton, M.; Lauritzan, S., Finite deFinetti Theorems in Linear Models and Multivariate Analysis. Scand. Jour. Statist. 1992, 19, 289-315. MR 94g:60065
38. Dyson, F., Statistical Theory of the Energy Levels of Complex Systems, I, II, III. J. Math. Phys. 1962, 3, 140-156, 157-165, 166-175. MR 26:1111, MR 26:1112, MR 26:1113
39. Dyson, F., Correlations Between Eigenvalues of a Random Matrix. Comm. Math. Phys. 1970, 19, 235-250. MR 43:4398
40. Eaton, M., Multivariate Statistics; Wiley: New York, 1983. MR 861:62086
41. Edelman, A.; Kostlan, E.; Shub, M., How Many Eigenvalues of a Random Matrix Are Real? Jour. Amer. Math. Soc. 1994, 7, 297-267. MR 94f:60053
42. Forrester, P.; Rains, E., Inter-Relationships Between Orthogonal, Unitary and Symplectic Matrix Ensembles. MSRI Publications 2001, 40, 171-207. MR 2002h:82008
43. Fulman, J., Random Matrix Theory Over Finite Fields. Bull. Amer. Math. Soc. 2002, 39, 51-86. MR 2002i:60012
44. Fulton, W., Eigenvalues, Invariant Factors, Highest Weights and Schubert Calculus. Bull. Amer. Math. Soc. 2000, 37, 209-249. MR 2001g:15023
45. Fyodorov, Y.; Khoruzhenko, B.; Sommers, H., Universality in the Random Matrix Spectra in the Regime of Weak Non-Hermiticity. Ann. Inst. Henri Poincaré: Physique Théorique 1998, 68, 440-489. MR 991:60080
46. Goldsheid, I.; Khoruzhenko, B., Eigenvalue Curves of Asymmetric Tri-Diagonal Random Matrices. Electronic Jour. Probab. 2000, 5, Paper 16. MR 2002j:82061
47. Goodman, R.; Wallach, W., Representations and Invariants of the Classical Groups, Cambridge Press: Cambridge, 1998. MR 99b:20073
48. Grenander, U.; Szegö, G., Toeplitz Forms and Their Applications. University of California Press: Berkeley, 1958. MR 20:1349
49. Haake, F., Secular Determinants of Random Unitary Matrices. Jour. Pys. A. 1996, 29, 3641-3658. MR 97g:82002
50. Haake, F., Quantum Signatures of Chaos, 2nd Ed.; Springer-Verlag: Berlin, 2001.
51. Hanlon, P.; Stanley, R.; Stembridge, J., Some Combinatorial Aspects of the Spectra of Normally Distributed Random Matrices. Contemp. Math. 1992, 138, 151-174. MR 93j:05164
52. Hirschman, I., The Strong Szegö Limit Theorem for Toeplitz Determinants. Amer. Jour. Math. 1966, 88, 577-614. MR 35:2064
53. Hughes, C.; Keating, J.; O'Connell, W., On the Characteristic Polynomial of a Random Unitary Matrix. Comm. Math. Phys. 2001, 220, 429-451. MR 2002m:82028
54. Hughes, C.; Rudnick, Z., Mock-Gaussian Behavior for Linear Statistics of Classical Compact Groups. Department of Mathematics, Tel Aviv University, 2002, preprint.
55. Johansson, K., On Szegö's Asymptotic Formula for Toeplitz Determinants and Generalizations. Bull. Sc. Math. 1988, 112, 257-304. MR 89m:47021
56. Johansson, K., On Random Matrices from the Compact Classical Groups. Ann. Math. 1997, 145, 519-545. MR 98e:60016
57. Johansson, K., The Longest Increasing Subsequence in a Random Permutation and a Unitary Random Matrix Model. Math. Res. Lett. 1998, 5, 63-82. MR 99e:60033
58. Johnstone, I., On the Distribution of the Largest Eigenvalue in Principal Component Analysis. Ann. Statist. 2001, 29, 295-327. MR 20021:62115
59. Katz, N.; Sarnak, P., Random Matrices, Frobenius Eigenvalues, and Monodromy. Amer. Math. Soc.: Providence, RI, 1999. MR 2000b:11070
60. Keating, J.; Snaith, N., Random Matrix Theory and ζ(1/2 + it). Commun. Math. Phys. 2000, 214, 57-89. MR 2002c:11107
61. Keating, J.; Snaith, N., Random Matrix Theory and L-Functions at s = 1/2. Commun. Math. Phys. 2000, 214, 91-110. MR 2002c:11108
62. Kiessling, M.; Spohn, H., A Note on the Eigenvalue Density of Random Matrices. Comm. Math. Phys. 1999, 199, 638-695. MR 2000a:82031
63. Macchi, O., Stochastic Processes and Multicoincidences. IEEE Transactions 1971, 17, 1-7.
64. Macchi, O., The Coincidence Approach to Stochastic Point Processes. Adv. Appl. Probab. 1975, 7, 83-122. MR 52:1876
65. MacDonald, I., Symmetric Functions and Hall Polynomials, 2nd Ed.; Clarendon Press: Oxford, 1995. MR 96h:05207
66. Marchenko, V.; Pastur, L., Distribution of Some Sets of Random Matrices. Mat. Sb. 1967, 1, 507-536.
67. Mardia, K.; Kent, J.; Bibby, J., Multivariate Analysis. Academic Press: New York, 1979. MR 81h:62003
68. Mehta, M., Random Matrices, 2nd Ed.; Acad. Press: New York, 1991. MR 92f:82002
69. Mezzadri, F., Random Matrix Theory and the Zeros of ζ′(s). Dept. of Mathematics, University of Bristol, 2002, preprint.
70. Muirhead, R., Latent Roots and Matrix Variates: A Review of Some Aymptotic Results. Ann. Statist. 1978, 6, 5-33. MR 56:16919
71. Odlyzko, A., On the Distribution of Spacings Between Zeros of the Zeta Function. Math. Comp. 1987, 48, 273-308. MR 88d:11082
72. 20-th Zero of the Riemann Zeta Function and 175 Million of Its Neighbors. ATT Laboratories, 1992, preprint.
73. O'Connell, N., Random Matrices, Non-Colliding Processes and Queues. Laboratoire de Probabilites, Paris 6, 2002, preprint.
74. O'Connell, N.; Yor, M., Brownian Analogues of Burke's Theorem. Stoch. Proc. Appl. 2001, 96, 285-304. MR 2002h:60175
75. Okounkov, A., Random Matrices and Random Permutations. Math. Res. Notices 2000, 20, 1043-1095. MR 2002c:15045
76. Olshansky, G., An Introduction to Harmonic Analysis on the Infinite-Dimensional Unitary Group. University of Pennsylvania, Dept. of Mathematics, 2001, preprint.
77. Olshanski, G.; Vershik, A., Ergodic Unitarily Invariant Measures on the Space of Infinite Hermitian Matrices. In Contemporary Mathematical Physics; Amer. Soc. Transl. Ser. 2, 1996, 175, 137-175. MR 98e:28015
78. Pickrell, D., Mackey Analysis of Infinite Classical Motion Groups. Pacific Jour. 1991, 150, 139-166. MR 92g:22041
79. Porod, U., The Cut-Off Phenomenon for Random Reflections. Ann. Probab. 1996, 24, 74-96. MR 97e:60012
80. Rains, E., High Powers of Random Elements of Compact Lie Groups. Probab. Th. Related Fields 107, 219-241. MR 98b:15026
81. Rains, E., Images of Eigenvalue Distributions Under Power Maps. ATT Laboratories, 1999, preprint.
82. Rains, E., Probability Theory on Compact Classical Groups., Harvard University: Department of Mathematics, 1991, Ph.D. thesis.
83. Rosenthal, J., Random Rotations, Characters and Random Walks on SO(N). Ann Probab. 1994, 22, 398-423. MR 95c:60008
84. Sinai, Y.; Soshnikov, A., Central Limit Theorem for Traces of Large Random Symmetric Matrices with Independent Matrix Elements. Bol. Soc. Brasil. Mat. (N.S.) 1998, 29, 1-24. MR 99f:60053
85. Sloane, N., Encrypting by Random Rotations. Technical Memorandum, Bell Laboratories, 1983.
86. Soshnikov, A., The Central Limit Theorem for Local Linear Statistics in Classical Compact Groups and Related Combinatorial Identities. Ann. Probab. 2000, 28, 1353-1370. MR 2002f:15035
87. Soshnikov, A., Level Spacings Distribution for Large Random Matrices: Gaussian Fluctuations. Ann. Math. 1998, 148, 573-617. MR 2000f:15014
88. Soshnikov, A., Determinantal Random Point Fields. Russian Math. Surveys 2000, 55, 923-975. MR 2002f:60097
89. Stanley, R., Enumerative Combinatorics. Vol. 2; Cambridge University Press: Cambridge, 1999. MR 2000k:05026
90. Tracy, C.; Widom, H., Introduction to Random Matrices. In Geometric and Quantum Aspects of Integrable Systems; Springer-Verlag: Berlin, 1993, 103-130. MR 95a:82050
91. Tracy, C.; Widom, H., Random Unitary Matrices, Permutations and Painlevé. Comm. Math. Physics 1999, 207, 665-685. MR 2001h:15019
92. Tracy, C.; Widom, H., On the Relations Between Orthogonal, Symplectic and Unitary Ensembles. Jour. Statist. Phys. 1999, 94, 347-363.
93. Tracy, C.; Widom, H., Universality of the Distribution Functions of Random Matrix Theory. CRM Proceedings 2000, 26, 251-264. MR 2002f:15036
94. Tracy, C.; Widom, H., On the Limit of Some Toeplitz-Like Determinants. SIAM J. Matrix Anal. Appl. 2002, 23, 1194-1196.
95. Voiculescu, D., Lectures on Free Probability Theory. Springer Lecture Notes in Mathematics 2000, 1738, 279-349. MR 2001g:46121
96. Wieand, K., Eigenvalue Distributions of Random Matrices in the Permutation Group and Compact Lie Groups, Harvard University: Department of Mathematics, 1998, Ph.D. thesis.
97. Wieand, K., Eigenvalue Distributions of Random Permutation Matrices. Ann. Probab. 2000, 28, 1563-1587. MR 2002d:15027