Truncations of Random Orthogonal Matrices

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong non--orthogonality, $M/N=$const, the behavior typical to real Ginibre ensemble is found. In the case $M=N-L$ with fixed $L$, a universal distribution of resonance widths is recovered.Comment: 4 pages, final revised version (one reference added, minor changes in Introduction

Lay Theories About White Racists: What Constitutes Racism (and What Doesn't)

Psychological theories of racial bias assume a pervasive motivation to avoid appearing racist, yet researchers know little regarding laypeople's theories about what constitutes racism. By investigating lay theories of White racism across both college and community samples, we seek to develop a more complete understanding of the nature of race-related norms, motivations, and processes of social perception in the contemporary United States. Factor analyses in Studies 1 and 1a indicated three factors underlying the traits laypeople associate with White racism: evaluative, psychological, and demographic. Studies 2 and 2a revealed a three-factor solution for behaviors associated with White racism: discomfort/unfamiliarity, overt racism, and denial of problem. For both traits and behaviors, lay theories varied by participants' race and their race-related attitudes and motivations. Specifically, support emerged for the prediction that lay theories of racism reflect a desire to distance the self from any aspect of the category ‘racist’

Hierarchical solutions of the Sherrington-Kirkpatrick model: Exact asymptotic behavior near the critical temperature

We analyze the replica-symmetry-breaking construction in the Sherrington-Kirkpatrick model of a spin glass. We present a general scheme for deriving an exact asymptotic behavior near the critical temperature of the solution with an arbitrary number of discrete hierarchies of the broken replica symmetry. We show that all solutions with finite-many hierarchies are unstable and only the scheme with infinite-many hierarchies becomes marginally stable. We show how the solutions from the discrete replica-symmetry-breaking scheme go over to the continuous one with increasing the number of hierarchies.Comment: REVTeX4, 11 pages, no figure

Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry

The statistical properties of quantum transport through a chaotic cavity are encoded in the traces \T={\rm Tr}(tt^\dag)^n, where $t$ is the transmission matrix. Within the Random Matrix Theory approach, these traces are random variables whose probability distribution depends on the symmetries of the system. For the case of broken time-reversal symmetry, we present explicit closed expressions for the average value and for the variance of \T for all $n$. In particular, this provides the charge cumulants \Q of all orders. We also compute the moments $$ of the conductance $g=\mathcal{T}_1$. All the results obtained are exact, {\it i.e.} they are valid for arbitrary numbers of open channels.Comment: 5 pages, 4 figures. v2-minor change

Analysis of the infinity-replica symmetry breaking solution of the Sherrington-Kirkpatrick model

In this work we analyse the Parisi's infinity-replica symmetry breaking solution of the Sherrington - Kirkpatrick model without external field using high order perturbative expansions. The predictions are compared with those obtained from the numerical solution of the infinity-replica symmetry breaking equations which are solved using a new pseudo-spectral code which allows for very accurate results. With this methods we are able to get more insight into the analytical properties of the solutions. We are also able to determine numerically the end-point x_{max} of the plateau of q(x) and find that lim_{T --> 0} x_{max}(T) > 0.5.Comment: 15 pages, 11 figures, RevTeX 4.

Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance

Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via M open channels. By using the supersymmetry method we derive: (i) an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane (ii) an explicit expression for the parametric correlation function of densities of eigenphases of the S-matrix. We use it to find the distribution of derivatives of these eigenphases with respect to the energy ("partial delay times" ) as well as with respect to an arbitrary external parameter.Comment: 51 pages, RevTEX , three figures are available on request. To be published in the special issue of the Journal of Mathematical Physic

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.Comment: 37 pages, 24 figure

Thermodynamic Properties and Phase Transitions in a Mean-Field Ising Spin Glass on Lattice Gas: the Random Blume-Emery-Griffiths-Capel Model

The study of the mean-field static solution of the Random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable Replica Symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a Full Replica Symmetry Breaking spin-glass phase. The nature of the transition can be either of the second order (like in the Sherrington-Kirkpatrick model) or, at temperature below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter and accompanied by a latent heat. In this last case coexistence of phases takes place. The thermodynamics is worked out in the Full Replica Symmetry Breaking scheme, and the relative Parisi equations are solved using a pseudo-spectral method down to zero temperature.Comment: 24 pages, 12 figure

Distribution of G-concurrence of random pure states

Average entanglement of random pure states of an N x N composite system is analyzed. We compute the average value of the determinant D of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distribution P(D). Similar results are obtained for the rescaled N-th root of the determinant, called G-concurrence. We show that in the limit $N\to\infty$ this quantity becomes concentrated at a single point G=1/e. The position of the concentration point changes if one consider an arbitrary N x K bipartite system, in the joint limit $N,K\to\infty$, K/N fixed.Comment: RevTeX4, 11 pages, 4 Encapsuled PostScript figures - Introduced new results, Section II and V have been significantly improved - To appear on PR

Calculation of the unitary part of the Bures measure for N-level quantum systems

We use the canonical coset parameterization and provide a formula with the unitary part of the Bures measure for non-degenerate systems in terms of the product of even Euclidean balls. This formula is shown to be consistent with the sampling of random states through the generation of random unitary matrices