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Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. (English) Zbl 1161.15302
Summary: 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.

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 15B52 Random matrices (algebraic aspects)
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##### References:
 [1] Adler, M.; Van Moerbeke, P., Integrals over Classical Groups, Random Permutations, Toda and Toeplitz Lattices. Comm. Pure Appl. Math. 2001, 54, 153-205. · Zbl 1086.34545 [2] Adler, M.; Van Moerbeke, P., Recursion Relations for Unitary Integrals, Combinatorics and the Toeplitz Lattice. Technical Report, Dept. of Mathematics, Brandeis, 2002. · Zbl 1090.37051 [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. · Zbl 1061.37047 [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. · Zbl 0937.60001 [5] T. W. Anderson, Asymptotic theory for principal component analysis, Ann. Math. Statist. 34 (1963), 122 – 148. · Zbl 0202.49504 [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. · Zbl 0949.60077 [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. · Zbl 0932.05001 [10] Estelle L. Basor, Distribution functions for random variables for ensembles of positive Hermitian matrices, Comm. Math. Phys. 188 (1997), no. 2, 327 – 350. · Zbl 0905.47016 [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. · Zbl 0948.47009 [12] M. V. Berry and J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev. 41 (1999), no. 2, 236 – 266. · Zbl 0928.11036 [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. [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. · Zbl 0916.15012 [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. · Zbl 1081.82569 [20] Daniel Bump and Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2002), no. 2, 252 – 271. · Zbl 1005.47030 [21] Bump, D.; Diaconis, P.; Keller, J., Unitary Correlations and the Fejer Kernel. Mathematical Phys., Analysis, Geometry 2002, 5, 101-123. · Zbl 1005.15014 [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. · Zbl 1075.11058 [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. · Zbl 1074.11046 [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. · Zbl 0657.60069 [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; American Mathematical Society, Providence, RI, 1999. · Zbl 0997.47033 [27] Percy Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), no. 6, 631 – 640. · Zbl 1041.05004 [28] Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes — Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. · Zbl 0695.60012 [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. · Zbl 0631.60018 [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. · Zbl 0586.60012 [31] Diaconis, P.; Shahshahani, M., The Subgroup Algorithm for Generating Uniform Random Variables. Prob. Eng. and Info. Sci. 1987, 1, 15-32. · Zbl 1133.60300 [32] Persi Diaconis and Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49 – 62. Studies in applied probability. · Zbl 0807.15015 [33] Persi Diaconis and Steven N. Evans, Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2615 – 2633. · Zbl 1008.15013 [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. · Zbl 0965.15007 [35] Diaconis, P.; Evans, S., A Different Construction of Gaussian Fields from Markov Chains: Dirichlet Covariances. Ann. Inst. Henri Poincaré, 2002, to appear. · Zbl 1033.60049 [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). · Zbl 0619.60039 [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. · Zbl 0795.62049 [38] Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys. 3 (1962), 140 – 156. , https://doi.org/10.1063/1.1703773 Freeman J. Dyson, Statistical theory of the energy levels of complex systems. II, J. Mathematical Phys. 3 (1962), 157 – 165. , https://doi.org/10.1063/1.1703774 Freeman J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys. 3 (1962), 166 – 175. · Zbl 0105.41604 [39] Freeman J. Dyson, Correlations between eigenvalues of a random matrix, Comm. Math. Phys. 19 (1970), 235 – 250. · Zbl 0221.62019 [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. · Zbl 0587.62097 [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. · Zbl 0790.15017 [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. · Zbl 0987.15004 [43] Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51 – 85. · Zbl 0984.60017 [44] William Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209 – 249. · Zbl 0994.15021 [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. · Zbl 0907.15017 [46] Ilya Ya. Goldsheid and Boris A. Khoruzhenko, Eigenvalue curves of asymmetric tridiagonal random matrices, Electron. J. Probab. 5 (2000), no. 16, 28. · Zbl 0983.82006 [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. · Zbl 0901.22001 [48] Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. · Zbl 0080.09501 [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. · Zbl 0899.15013 [50] Haake, F., Quantum Signatures of Chaos, 2nd Ed.; Springer-Verlag: Berlin, 2001. · Zbl 0985.81038 [51] Philip J. Hanlon, Richard P. Stanley, and John R. Stembridge, Some combinatorial aspects of the spectra of normally distributed random matrices, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 151 – 174. · Zbl 0789.05092 [52] I. I. Hirschman Jr., The strong Szegö limit theorem for Toeplitz determinants, Amer. J. Math. 88 (1966), 577 – 614. · Zbl 0173.42602 [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. · Zbl 0987.60039 [54] Hughes, C.; Rudnick, Z., Mock-Gaussian Behavior for Linear Statistics of Classical Compact Groups. Department of Mathematics, Tel Aviv University, 2002, preprint. · Zbl 1034.60003 [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). · Zbl 0661.30001 [56] Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (1997), no. 3, 519 – 545. · Zbl 0883.60010 [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. · Zbl 0923.60008 [58] Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), no. 2, 295 – 327. · Zbl 1016.62078 [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. · Zbl 0958.11004 [60] J. P. Keating and N. C. Snaith, Random matrix theory and \?(1/2+\?\?), Comm. Math. Phys. 214 (2000), no. 1, 57 – 89. · Zbl 1051.11048 [61] J. P. Keating and N. C. Snaith, Random matrix theory and \?-functions at \?=1/2, Comm. Math. Phys. 214 (2000), no. 1, 91 – 110. · Zbl 1051.11047 [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. · Zbl 0928.15015 [63] Macchi, O., Stochastic Processes and Multicoincidences. IEEE Transactions 1971, 17, 1-7. · Zbl 0221.60075 [64] Odile Macchi, The coincidence approach to stochastic point processes, Advances in Appl. Probability 7 (1975), 83 – 122. · Zbl 0366.60081 [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. · Zbl 0899.05068 [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-New York-Toronto, Ont., 1979. Probability and Mathematical Statistics: A Series of Monographs and Textbooks. [68] Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. · Zbl 0780.60014 [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. · Zbl 0375.62050 [71] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273 – 308. · Zbl 0615.10049 [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. · Zbl 1058.60078 [75] Andrei Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 20 (2000), 1043 – 1095. · Zbl 1018.15020 [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. [78] Doug Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math. 150 (1991), no. 1, 139 – 166. · Zbl 0739.22016 [79] Ursula Porod, The cut-off phenomenon for random reflections, Ann. Probab. 24 (1996), no. 1, 74 – 96. · Zbl 0854.60068 [80] E. M. Rains, High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107 (1997), no. 2, 219 – 241. · Zbl 0868.60012 [81] Rains, E., Images of Eigenvalue Distributions Under Power Maps. ATT Laboratories, 1999, preprint. · Zbl 1068.60014 [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 \?\?(\?), Ann. Probab. 22 (1994), no. 1, 398 – 423. · Zbl 0799.60007 [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. · Zbl 0912.15027 [85] Sloane, N., Encrypting by Random Rotations. Technical Memorandum, Bell Laboratories, 1983. · Zbl 0507.94010 [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. · Zbl 1021.60018 [87] Alexander Soshnikov, Level spacings distribution for large random matrices: Gaussian fluctuations, Ann. of Math. (2) 148 (1998), no. 2, 573 – 617. · Zbl 0944.60060 [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. · Zbl 0991.60038 [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. · Zbl 0928.05001 [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. · Zbl 0791.15017 [91] Craig A. Tracy and Harold Widom, Random unitary matrices, permutations and Painlevé, Comm. Math. Phys. 207 (1999), no. 3, 665 – 685. · Zbl 0965.60028 [92] Tracy, C.; Widom, H., On the Relations Between Orthogonal, Symplectic and Unitary Ensembles. Jour. Statist. Phys. 1999, 94, 347-363. · Zbl 0935.60090 [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. · Zbl 0973.60057 [94] Tracy, C.; Widom, H., On the Limit of Some Toeplitz-Like Determinants. SIAM J. Matrix Anal. Appl. 2002, 23, 1194-1196. · Zbl 1010.47019 [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. · Zbl 0941.00026 [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. · Zbl 1044.15017
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