Exchangeable pairs and Poisson approximation

This is a survey paper on Poisson approximation using Stein's method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector's problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.Comment: Published at http://dx.doi.org/10.1214/154957805100000096 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cosmological baryonic and matter densities from 600,000 SDSS Luminous Red Galaxies with photometric redshifts

We analyze MegaZ-LRG, a photometric-redshift catalogue of Luminous Red Galaxies (LRGs) based on the imaging data of the Sloan Digital Sky Survey (SDSS) 4th Data Release. MegaZ-LRG, presented in a companion paper, contains 10^6 photometric redshifts derived with ANNz, an Artificial Neural Network method, constrained by a spectroscopic sub-sample of 13,000 galaxies obtained by the 2dF-SDSS LRG and Quasar (2SLAQ) survey. The catalogue spans the redshift range 0.4 < z < 0.7 with an r.m.s. redshift error ~ 0.03(1+z), covering 5,914 deg^2 to map out a total cosmic volume 2.5 h^-3 Gpc^3. In this study we use the most reliable 600,000 photometric redshifts to present the first cosmological parameter fits to galaxy angular power spectra from a photometric redshift survey. Combining the redshift slices with appropriate covariances, we determine best-fitting values for the matter and baryon densities of Omega_m h = 0.195 +/- 0.023 and Omega_b/Omega_m = 0.16 +/- 0.036 (with the Hubble parameter h = 0.75 and scalar index of primordial fluctuations n = 1 held fixed). These results are in agreement with and independent of the latest studies of the Cosmic Microwave Background radiation, and their precision is comparable to analyses of contemporary spectroscopic-redshift surveys. We perform an extensive series of tests which conclude that our power spectrum measurements are robust against potential systematic photometric errors in the catalogue. We conclude that photometric-redshift surveys are competitive with spectroscopic surveys for measuring cosmological parameters in the simplest vanilla models. Future deep imaging surveys have great potential for further improvement, provided that systematic errors can be controlled.Comment: 24 pages, 23 figures, MNRAS accepte

Random matrices, random polynomials and Coulomb systems

It is well known that the joint probability density of the eigenvalues of Gaussian ensembles of random matrices may be interpreted as a Coulomb gas. We review these classical results for hermitian and complex random matrices, with special attention devoted to electrostatic analogies. We also discuss the joint probability density of the zeros of polynomials whose coefficients are complex Gaussian variables. This leads to a new two-dimensional solvable gas of interacting particles, with non-trivial interactions between particles.Comment: 8 pages, to appear in the Proceedings of the International Conference on Strongly Coupled Coulomb Systems, Saint-Malo, 199

Probability in Orthodox Quantum Mechanics: Probability as a Postulate Versus Probability as an Emergent Phenomenon

The role of probability in quantum mechanics is reviewed, with a discussion of the ``orthodox'' versus the statistical interpretive frameworks, and of a number of related issues. After a brief summary of sources of unease with quantum mechanics, a survey is given of attempts either to give a new interpretive framework assuming quantum mechanics is exact, or to modify quantum mechanics assuming it is a very accurate approximation to a more fundamental theory. This survey focuses particularly on the issues of whether probabilities in quantum mechanics are postulated or emergent.Comment: Latex; Submitted to the Proceedings of the Ischia Conference on ``Chance in Physics: Foundations and Perspectives'