Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

Diaconis, Persi

doi:10.1090/S0273-0979-03-00975-3

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-9485(online) ISSN 0273-0979(print)

Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture

Author: Persi Diaconis
Journal: Bull. Amer. Math. Soc. 40 (2003), 155-178
MSC (2000): Primary 00-02, 60B15
Posted: February 12, 2003
MathSciNet review: 1962294
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Typical large unitary matrices show remarkable patterns in their eigenvalue distribution. These same patterns appear in telephone encryption, the zeros of Riemann's zeta function, a variety of physics problems, and in the study of Toeplitz operators. This paper surveys these applications and what is currently known about the patterns.

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. M. Adler, T. Shiota, and P. van Moerbeke, Random matrices, Virasoro algebras, and noncommutative KP, Duke Math. J. 94 (1998), no. 2, 379–431. MR 1638599 (99e:58088), http://dx.doi.org/10.1215/S0012-7094-98-09417-0
4. David Aldous and Persi Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 413–432. MR 1694204 (2000g:60013), http://dx.doi.org/10.1090/S0273-0979-99-00796-X
5. T. W. Anderson, Asymptotic theory for principal component analysis, Ann. Math. Statist. 34 (1963), 122–148. MR 0145620 (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. Z. D. Bai, Methodologies in spectral analysis of large-dimensional random matrices, a review, Statist. Sinica 9 (1999), no. 3, 611–677. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. MR 1711663 (2000e:60044)
9. Jinho Baik, Percy Deift, and Kurt 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 1682248 (2000e:05006), http://dx.doi.org/10.1090/S0894-0347-99-00307-0
10. Estelle L. Basor, Distribution functions for random variables for ensembles of positive Hermitian matrices, Comm. Math. Phys. 188 (1997), no. 2, 327–350. MR 1471817 (99b:82046), http://dx.doi.org/10.1007/s002200050167
11. Estelle L. Basor, Connections between random matrices and Szegö limit theorems, Spectral problems in geometry and arithmetic (Iowa City, IA, 1997), Contemp. Math., vol. 237, Amer. Math. Soc., Providence, RI, 1999, pp. 1–7. MR 1710785 (2000e:47049), http://dx.doi.org/10.1090/conm/237/1710785
12. M. V. Berry and J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev. 41 (1999), no. 2, 236–266 (electronic). MR 1684543 (2000f:11107), http://dx.doi.org/10.1137/S0036144598347497
13. Biane, P., Free Probability for Probabilists, 2000, preprint.
14. Oriol Bohigas and Marie-Joya Giannoni, Chaotic motion and random matrix theories, Mathematical and computational methods in nuclear physics (Granada, 1983), Lecture Notes in Phys., vol. 209, Springer, Berlin, 1984, pp. 1–99. MR 769113 (86c:58129), http://dx.doi.org/10.1007/3-540-13392-5_1
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. Albrecht Böttcher and Bernd Silbermann, Introduction to large truncated Toeplitz matrices, Universitext, Springer-Verlag, New York, 1999. MR 1724795 (2001b:47043)
18. Bougerol, Ph. and Jeulin, Th., Paths in Weyl Chambers and Random Matrices. Laboratoire de Probabilities: Paris 2001, preprint.
19. A. Boutet de Monvel, L. Pastur, and M. Shcherbina, On the statistical mechanics approach in the random matrix theory: integrated density of states, J. Statist. Phys. 79 (1995), no. 3-4, 585–611. MR 1327898 (96d:82033), http://dx.doi.org/10.1007/BF02184872
20. Daniel Bump and Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2002), no. 2, 252–271. MR 1883866 (2002j:47052), http://dx.doi.org/10.1006/jcta.2001.3214
21. Bump, D.; Diaconis, P.; Keller, J., Unitary Correlations and the Fejer Kernel. Mathematical Phys., Analysis, Geometry 2002, 5, 101-123.
22. Conrey, B., -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. D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York, 1988. MR 950166 (90e:60060)
26. P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University Courant Institute of Mathematical Sciences, New York, 1999. MR 1677884 (2000g:47048)
27. Percy Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), no. 6, 631–640. MR 1764262 (2001g:05012)
28. Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 11, Institute of Mathematical Statistics, Hayward, CA, 1988. MR 964069 (90a:60001)
29. Persi Diaconis, Application of the method of moments in probability and statistics, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 125–142. MR 921087 (89m:60006)
30. Persi Diaconis and Mehrdad Shahshahani, Products of random matrices as they arise in the study of random walks on groups, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 183–195. MR 841092 (87k:60025), http://dx.doi.org/10.1090/conm/050/841092
31. Diaconis, P.; Shahshahani, M., The Subgroup Algorithm for Generating Uniform Random Variables. Prob. Eng. and Info. Sci. 1987, 1, 15-32.
32. Persi Diaconis and Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49–62. Studies in applied probability. MR 1274717 (95m:60011)
33. Persi Diaconis and Steven N. Evans, Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2615–2633. MR 1828463 (2002d:60003), http://dx.doi.org/10.1090/S0002-9947-01-02800-8
34. Persi Diaconis and Steven N. Evans, Immanants and finite point processes, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 305–321. In memory of Gian-Carlo Rota. MR 1780025 (2001m:15018), http://dx.doi.org/10.1006/jcta.2000.3097
35. Diaconis, P.; Evans, S., A Different Construction of Gaussian Fields from Markov Chains: Dirichlet Covariances. Ann. Inst. Henri Poincaré, 2002, to appear.
36. Persi Diaconis and David Freedman, A dozen de Finetti-style results in search of a theory, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 397–423 (English, with French summary). MR 898502 (88f:60072)
37. Persi W. Diaconis, Morris L. Eaton, and Steffen L. Lauritzen, Finite de Finetti theorems in linear models and multivariate analysis, Scand. J. Statist. 19 (1992), no. 4, 289–315. MR 1211786 (94g:60065)
38. Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140–156. MR 0143556 (26 #1111)
Freeman J. Dyson, Statistical theory of the energy levels of complex systems. II, J. Mathematical Phys. 3 (1962), 157–165. MR 0143557 (26 #1112)
Freeman J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys. 3 (1962), 166–175. MR 0143558 (26 #1113)
39. Freeman J. Dyson, Correlations between eigenvalues of a random matrix, Comm. Math. Phys. 19 (1970), 235–250. MR 0278668 (43 #4398)
40. Morris L. Eaton, Multivariate statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1983. A vector space approach. MR 716321 (86i:62086)
41. Alan Edelman, Eric Kostlan, and Michael Shub, How many eigenvalues of a random matrix are real?, J. Amer. Math. Soc. 7 (1994), no. 1, 247–267. MR 1231689 (94f:60053), http://dx.doi.org/10.1090/S0894-0347-1994-1231689-0
42. Peter J. Forrester and Eric M. Rains, Interrelationships between orthogonal, unitary and symplectic matrix ensembles, Random matrix models and their applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 171–207. MR 1842786 (2002h:82008)
43. Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51–85. MR 1864086 (2002i:60012), http://dx.doi.org/10.1090/S0273-0979-01-00920-X
44. William Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249 (electronic). MR 1754641 (2001g:15023), http://dx.doi.org/10.1090/S0273-0979-00-00865-X
45. Yan V. Fyodorov, Hans-Jürgen Sommers, and Boris A. Khoruzhenko, Universality in the random matrix spectra in the regime of weak non-Hermiticity, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 4, 449–489 (English, with English and French summaries). Classical and quantum chaos. MR 1634312 (99i:60080)
46. Ilya Ya. Goldsheid and Boris A. Khoruzhenko, Eigenvalue curves of asymmetric tridiagonal random matrices, Electron. J. Probab. 5 (2000), no. 16, 28 pp. (electronic). MR 1800072 (2002j:82061), http://dx.doi.org/10.1214/EJP.v5-72
47. Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831 (99b:20073)
48. Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, 1958. MR 0094840 (20 #1349)
49. Fritz Haake, Marek Kuś, Hans-Jürgen Sommers, Henning Schomerus, and Karol Życzkowski, Secular determinants of random unitary matrices, J. Phys. A 29 (1996), no. 13, 3641–3658. MR 1400168 (97g:82002), http://dx.doi.org/10.1088/0305-4470/29/13/029
50. Haake, F., Quantum Signatures of Chaos, 2nd Ed.; Springer-Verlag: Berlin, 2001.
51. Philip J. Hanlon, Richard P. Stanley, and John R. Stembridge, Some combinatorial aspects of the spectra of normally distributed random matrices, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 151–174. MR 1199126 (93j:05164), http://dx.doi.org/10.1090/conm/138/1199126
52. I. I. Hirschman Jr., The strong Szegö limit theorem for Toeplitz determinants, Amer. J. Math. 88 (1966), 577–614. MR 0211182 (35 #2064)
53. C. P. Hughes, J. P. Keating, and Neil O’Connell, On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 220 (2001), no. 2, 429–451. MR 1844632 (2002m:82028), http://dx.doi.org/10.1007/s002200100453
54. Hughes, C.; Rudnick, Z., Mock-Gaussian Behavior for Linear Statistics of Classical Compact Groups. Department of Mathematics, Tel Aviv University, 2002, preprint.
55. Kurt Johansson, On Szegő’s asymptotic formula for Toeplitz determinants and generalizations, Bull. Sci. Math. (2) 112 (1988), no. 3, 257–304 (English, with French summary). MR 975365 (89m:47021)
56. Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (1997), no. 3, 519–545. MR 1454702 (98e:60016), http://dx.doi.org/10.2307/2951843
57. Kurt Johansson, The longest increasing subsequence in a random permutation and a unitary random matrix model, Math. Res. Lett. 5 (1998), no. 1-2, 63–82. MR 1618351 (99e:60033)
58. Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), no. 2, 295–327. MR 1863961 (2002i:62115), http://dx.doi.org/10.1214/aos/1009210544
59. Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828 (2000b:11070)
60. J. P. Keating and N. C. Snaith, Random matrix theory and 𝜁(1/2+𝑖𝑡), Comm. Math. Phys. 214 (2000), no. 1, 57–89. MR 1794265 (2002c:11107), http://dx.doi.org/10.1007/s002200000261
61. J. P. Keating and N. C. Snaith, Random matrix theory and 𝐿-functions at 𝑠=1/2, Comm. Math. Phys. 214 (2000), no. 1, 91–110. MR 1794267 (2002c:11108), http://dx.doi.org/10.1007/s002200000262
62. Michael K.-H. Kiessling and Herbert Spohn, A note on the eigenvalue density of random matrices, Comm. Math. Phys. 199 (1999), no. 3, 683–695. MR 1669669 (2000a:82031), http://dx.doi.org/10.1007/s002200050516
63. Macchi, O., Stochastic Processes and Multicoincidences. IEEE Transactions 1971, 17, 1-7.
64. Odile Macchi, The coincidence approach to stochastic point processes, Advances in Appl. Probability 7 (1975), 83–122. MR 0380979 (52 #1876)
65. I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
66. Marchenko, V.; Pastur, L., Distribution of Some Sets of Random Matrices. Mat. Sb. 1967, 1, 507-536.
67. Kantilal Varichand Mardia, John T. Kent, and John M. Bibby, Multivariate analysis, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1979. Probability and Mathematical Statistics: A Series of Monographs and Textbooks. MR 560319 (81h:62003)
68. Madan Lal Mehta, Random matrices, 2nd ed., Academic Press Inc., Boston, MA, 1991. MR 1083764 (92f:82002)
69. Mezzadri, F., Random Matrix Theory and the Zeros of $\xi^\prime (s)$ . Dept. of Mathematics, University of Bristol, 2002, preprint.
70. Robb J. Muirhead, Latent roots and matrix variates: a review of some asymptotic results, Ann. Statist. 6 (1978), no. 1, 5–33. MR 0458719 (56 #16919)
71. A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308. MR 866115 (88d:11082), http://dx.doi.org/10.1090/S0025-5718-1987-0866115-0
72. Odlyzko, A., The $10^{20}$ -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. Neil O’Connell and Marc Yor, Brownian analogues of Burke’s theorem, Stochastic Process. Appl. 96 (2001), no. 2, 285–304. MR 1865759 (2002h:60175), http://dx.doi.org/10.1016/S0304-4149(01)00119-3
75. Andrei Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 20 (2000), 1043–1095. MR 1802530 (2002c:15045), http://dx.doi.org/10.1155/S1073792800000532
76. Olshansky, G., An Introduction to Harmonic Analysis on the Infinite-Dimensional Unitary Group. University of Pennsylvania, Dept. of Mathematics, 2001, preprint.
77. Grigori Olshanski and Anatoli Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, Contemporary mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 175, Amer. Math. Soc., Providence, RI, 1996, pp. 137–175. MR 1402920 (98e:28015)
78. Doug Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math. 150 (1991), no. 1, 139–166. MR 1120717 (92g:22041)
79. Ursula Porod, The cut-off phenomenon for random reflections, Ann. Probab. 24 (1996), no. 1, 74–96. MR 1387627 (97e:60012), http://dx.doi.org/10.1214/aop/1042644708
80. E. M. Rains, High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107 (1997), no. 2, 219–241. MR 1431220 (98b:15026), http://dx.doi.org/10.1007/s004400050084
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. Jeffrey S. Rosenthal, Random rotations: characters and random walks on 𝑆𝑂(𝑁), Ann. Probab. 22 (1994), no. 1, 398–423. MR 1258882 (95c:60008)
84. Ya. Sinai and A. Soshnikov, Central limit theorem for traces of large random symmetric matrices with independent matrix elements, Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), no. 1, 1–24. MR 1620151 (99f:60053), http://dx.doi.org/10.1007/BF01245866
85. Sloane, N., Encrypting by Random Rotations. Technical Memorandum, Bell Laboratories, 1983.
86. Alexander Soshnikov, The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), no. 3, 1353–1370. MR 1797877 (2002f:15035), http://dx.doi.org/10.1214/aop/1019160338
87. Alexander Soshnikov, Level spacings distribution for large random matrices: Gaussian fluctuations, Ann. of Math. (2) 148 (1998), no. 2, 573–617. MR 1668559 (2000f:15014), http://dx.doi.org/10.2307/121004
88. A. Soshnikov, Determinantal random point fields, Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 5, 923–975. MR 1799012 (2002f:60097), http://dx.doi.org/10.1070/rm2000v055n05ABEH000321
89. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282 (2000k:05026)
90. Craig A. Tracy and Harold Widom, Introduction to random matrices, Geometric and quantum aspects of integrable systems (Scheveningen, 1992), Lecture Notes in Phys., vol. 424, Springer, Berlin, 1993, pp. 103–130. MR 1253763 (95a:82050), http://dx.doi.org/10.1007/BFb0021444
91. Craig A. Tracy and Harold Widom, Random unitary matrices, permutations and Painlevé, Comm. Math. Phys. 207 (1999), no. 3, 665–685. MR 1727236 (2001h:15019), http://dx.doi.org/10.1007/s002200050741
92. Tracy, C.; Widom, H., On the Relations Between Orthogonal, Symplectic and Unitary Ensembles. Jour. Statist. Phys. 1999, 94, 347-363.
93. Craig A. Tracy and Harold Widom, Universality of the distribution functions of random matrix theory, Integrable systems: from classical to quantum (Montréal, QC, 1999), CRM Proc. Lecture Notes, vol. 26, Amer. Math. Soc., Providence, RI, 2000, pp. 251–264. MR 1791893 (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. Dan Voiculescu, Lectures on free probability theory, Lectures on probability theory and statistics (Saint-Flour, 1998) Lecture Notes in Math., vol. 1738, Springer, Berlin, 2000, pp. 279–349. MR 1775641 (2001g:46121), http://dx.doi.org/10.1007/BFb0106703
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. Kelly Wieand, Eigenvalue distributions of random permutation matrices, Ann. Probab. 28 (2000), no. 4, 1563–1587. MR 1813834 (2002d:15027), http://dx.doi.org/10.1214/aop/1019160498

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|