Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and derive the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.Comment: 46 pages, 13 figures, second revision adds explanations, figures, and reference

Statistical eigen-inference from large Wishart matrices

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., "eigen-inference"). Results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix. We develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations. We exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue-based procedures for testing and estimation. Specifically, we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the (nonrandom) population eigenvalues has a staircase structure. Monte Carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high-dimensional, (relatively) small sample size settings. The improved performance results from the fact that the proposed inference procedures are "global" (in a sense that we describe) and exploit "global" information thereby overcoming the inherent biases that cripple classical inference procedures which are "local" and rely on "local" information.Comment: Published in at http://dx.doi.org/10.1214/07-AOS583 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sharp Bounds for Sums Associated to Graphs of Matrices

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph G associated to the constraints in the sum.Comment: 20 page

Predicting Residential Satisfaction: A Comparative Case Study

This is a comparative case study that focuses on resident satisfaction in three buildings renovated for housing. A survey based on environment-behavior factors that can contribute to resident satisfaction was developed and distributed to the buildings\u27 residents. Residents in fifty-two percent (52.5%) of the units in the three buildings responded (N = 64). Index variables used were: management, perception, wayfinding, safety. comfort. and adequacy. There was a significant relationship between resident satisfaction and age for one building. Safety and perception were significant for all buildings. Safety, perception and comfort were significant in different ways for each of the three buildings

Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of "second order freeness", which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large $N$ limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.Comment: 31 pages, new section on failure of universality added, typos corrected, additional explanation

Relativistic Point-Coupling Models as Effective Theories of Nuclei

Recent studies have shown that concepts of effective field theory such as naturalness can be profitably applied to relativistic mean-field models of nuclei. Here the analysis by Friar, Madland, and Lynn of naturalness in a relativistic point-coupling model is extended. Fits to experimental nuclear data support naive dimensional analysis as a useful principle and imply a mean-field expansion analogous to that found for mean-field meson models.Comment: 26 pages, REVTeX 3.0 with epsf.sty, plus 5 figure

The neurovirulence and neuroinvasiveness of chimeric tick-borne encephalitis/dengue virus can be attenuated by introducing defined mutations into the envelope and NS5 protein genes and the 3′ non-coding region of the genome

AbstractTick-borne encephalitis (TBE) is a severe disease affecting thousands of people throughout Eurasia. Despite the use of formalin-inactivated vaccines in endemic areas, an increasing incidence of TBE emphasizes the need for an alternative vaccine that will induce a more durable immunity against TBE virus (TBEV). The chimeric attenuated virus vaccine candidate containing the structural protein genes of TBEV on a dengue virus genetic background (TBEV/DEN4) retains a high level of neurovirulence in both mice and monkeys. Therefore, attenuating mutations were introduced into the envelope (E315) and NS5 (NS5654,655) proteins, and into the 3′ non-coding region (Δ30) of TBEV/DEN4. The variant that contained all three mutations (vΔ30/E315/NS5654,655) was significantly attenuated for neuroinvasiveness and neurovirulence and displayed a reduced level of replication and virus-induced histopathology in the brains of mice. The high level of safety in the central nervous system indicates that vΔ30/E315/NS5654,655 should be further evaluated as a TBEV vaccine