Random Matrices in Physics

Abstract

Next article Random Matrices in PhysicsEugene P. WignerEugene P. Wignerhttps://doi.org/10.1137/1009001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] C. E. Porter, C. E. Porter, Fluctuations of quantal spectraStatistical Theories of Spectra: Fluctuations, Academic Press, New York, 1965 Google Scholar[2] Madan Lal Mehta, Random matrices, Academic Press Inc., Boston, MA, 1991xviii+562 MR1083764 0780.60014 Google Scholar[3] J. Wishart, Generalized product moment distribution in samples, Biometrika, 20 A (1928), 32–52, (Correction: Ibid., 20 A (1928), p. 425.) CrossrefGoogle Scholar[4] R. A. Fisher, The sampling distribution of some statistics obtained from non-linear equations, Ann. Eugenics, 9 (1939), 238–249, (now Ann. Human Genetics) MR0001499 0022.37202 CrossrefGoogle Scholar[5] A. T. James, Normal multivariate analysis and the orthogonal group, Ann. Math. Statistics, 25 (1954), 40–75 MR0060779 0055.13203 CrossrefISIGoogle Scholar[6] Samuel S. Wilks, Mathematical statistics, A Wiley Publication in Mathematical Statistics, John Wiley & Sons Inc., New York, 1962xvi+644, Chap. XI MR0144404 0173.45805 Google Scholar[7] Harald Cramér, Mathematical Methods of Statistics, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, N. J., 1946xvi+575 MR0016588 0063.01014 Google Scholar[8A] Freeman J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Mathematical Phys., 3 (1962), 140–156 10.1063/1.1703773 MR0143556 0105.41604 CrossrefISIGoogle Scholar[8B] Freeman J. Dyson, Statistical theory of the energy levels of complex systems. II, J. Mathematical Phys., 3 (1962), 157–165 10.1063/1.1703774 MR0143557 0105.41604 CrossrefISIGoogle Scholar[8C] Freeman J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys., 3 (1962), 166–175 10.1063/1.1703775 MR0143558 0105.41604 CrossrefISIGoogle Scholar[9] B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 17 (1931), 315–318 0002.05701 CrossrefGoogle Scholar[10] J. Von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. U.S.A., 18 (1932), 70–82 0004.31004 CrossrefGoogle Scholar[11] G. D. Birkhoff, Proof of a recurrence theorem for strongly transitive systems, Proc. Nat. Acad. Sci. U.S.A., 17 (1931), 650–655 0003.25601 CrossrefGoogle Scholar[12] G. D. Birkhoff, Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A., 17 (1931), 656–660 0003.25602 CrossrefGoogle Scholar[13] G. D. Birkhoff and , B. O. Koopman, Recent contributions to the ergodic theory, Proc. Nat. Acad. Sci. U.S.A., 18 (1932), 279–282 0004.31102 CrossrefGoogle Scholar[14] Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2), 62 (1955), 548–564 MR0077805 0067.08403 CrossrefISIGoogle Scholar[15] Eugene P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. (2), 67 (1958), 325–327 MR0095527 0085.13203 CrossrefISIGoogle Scholar[16] Mark Kac, Probability and related topics in physical sciences, With special lectures by G. E. Uhlenbeck, A. R. Hibbs, and B. van der Pol. Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, Colo., Vol. 1957, Interscience Publishers, London-New York, 1959, 11– MR0102849 0087.33003 Google Scholar[17] Burt V. Bronk, Accuracy of the semicircle approximation for the density of eigenvalues of random matrices, J. Mathematical Phys., 5 (1964), 215–220, (Also Princeton dissertation, 1963.) 10.1063/1.1704111 MR0161680 CrossrefISIGoogle Scholar[18] P. L. Hsu, On the distribution of roots of certain determinantal equations, Ann. Eugenics, 9 (1939), 250–258, (now Ann. Human Genetics) MR0001500 0022.24502 CrossrefGoogle Scholar[19] D. N. Nanda, Distribution of a root of a determinantal equation, Ann. Math. Statistics, 19 (1948), 47–57 MR0024106 0041.44903 CrossrefISIGoogle Scholar[20] Charles E. Porter and , Norbert Rosenzweig, Statistical properties of atomic and nuclear spectra, Ann. Acad. Sci. Fenn. Ser. A VI No., 44 (1960), 66–, (Also reprinted in [1] above.) MR0118377 0092.23304 Google Scholar[21] J. Von Neumann, Charakterisierung des Spectrums eines Integraloperators, Actualités Sci. Indust. (229), (1935), , (Reprinted in John von Neumann Collected Works, vol. IV, A. H. Taub, ed., Pergamon Press, New York, 1962.) Google Scholar[22] J. M. Blatt and , V. F. Weisskopf, Theoretical Nuclear Physics, John Wiley, New York, 1952, VIII-6 0049.14005 Google Scholar[23] T. D. Newton, Shell effects on the spacing of nuclear levels, Canad. J. Phys, 34 (1956), 804–829 0071.43106 CrossrefISIGoogle Scholar[24] Jean Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Mathematical Phys., 6 (1965), 440–449 10.1063/1.1704292 MR0173726 0127.39304 CrossrefISIGoogle Scholar[25] Eugene P. Wigner, On a class of analytic functions from the quantum theory of collisions, Ann. of Math. (2), 53 (1951)