We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis"--a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return.

We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. We find that C has the universal properties of the Gaussian orthogonal ensemble of random matrices. Furthermore, we analyze the eigenvectors of C through their inverse participation ratio and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum-a situation reminiscent of results in localization theory.

We show that the particle-hole conjugate of the Pfaffian state-or "anti-Pfaffian" state-is in a different universality class from the Pfaffian state, with different topological order. The two states can be distinguished easily by their edge physics: their edges differ in both their thermal Hall conductance and their tunneling exponents. At the same time, the two states are exactly degenerate in energy for a nu=5/2 quantum Hall system in the idealized limit of zero Landau level mixing. Thus, both are good candidates for the observed sigma_{xy}=5/2(e;{2}/h) quantum Hall plateau.

We analyze interference phenomena in the quantum-Hall analog of the Fabry-Perot interferometer, exploring the roles of the Aharonov-Bohm effect, Coulomb interactions, and fractional statistics on the oscillations of the resistance as one varies the magnetic field B and/or the voltage VG applied to a side gate. Coulomb interactions couple the interfering edge mode to localized quasiparticle states in the bulk, whose occupation is quantized in integer values. For the integer quantum Hall effect, if the bulk-edge coupling is absent, the resistance exhibits an Aharonov-Bohm (AB) periodicity, where the phase is equal to the number of quanta of magnetic flux enclosed by a specified interferometer area. When bulk-edge coupling is present, the actual area of the interferometer oscillates as function of B and VG, with a combination of a smooth variation and abrupt jumps due to changes in the number of quasi-particles in the bulk of the interferometer. This modulates the Aharonov-Bohm phase and gives rise to additional periodicities in the resistance. In the limit of strong interactions, the amplitude of the AB oscillations becomes negligible, and one sees only the new "Coulomb-dominated" (CD) periodicity. In the limits where either the AB or the CD periodicities dominate, a color map of resistance will show a series of parallel stripes in the B − VG plane, but the two cases show different stripe spacings and slopes of opposite signs. At intermediate coupling, one sees a superposition of the two patterns. We discuss dependences of the interference intensities on parameters including the temperature and the backscattering strengths of the individual constrictions. We also discuss how results are modified in a fractional quantized Hall system, and the extent to which the interferometer may demonstrate the fractional statistics of the quasiparticles.