Probability density of determinants of random matrices

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

Heterotic free fermionic and symmetric toroidal orbifold models

Free fermionic models and symmetric heterotic toroidal orbifolds both constitute exact backgrounds that can be used effectively for phenomenological explorations within string theory. Even though it is widely believed that for Z2xZ2 orbifolds the two descriptions should be equivalent, a detailed dictionary between both formulations is still lacking. This paper aims to fill this gap: We give a detailed account of how the input data of both descriptions can be related to each other. In particular, we show that the generalized GSO phases of the free fermionic model correspond to generalized torsion phases used in orbifold model building. We illustrate our translation methods by providing free fermionic realizations for all Z2xZ2 orbifold geometries in six dimensions.Comment: 1+49 pages latex, minor revisions and references adde

Random matrix models with log-singular level confinement: method of fictitious fermions

Joint distribution function of N eigenvalues of U(N) invariant random-matrix ensemble can be interpreted as a probability density to find N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic, log-singular level confinement. An effective one-particle Schroedinger equation for wave-functions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson's density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk, and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.Comment: 13 pages (latex), Presented at the MINERVA Workshop on Mesoscopics, Fractals and Neural Networks, Eilat, Israel, March 199

Magnetic susceptibility of ultra-small superconductor grains

For assemblies of superconductor nanograins, the magnetic response is analyzed as a function of both temperature and magnetic field. In order to describe the interaction energy of electron pairs for a huge number of many-particle states, involved in calculations, we develop a simple approximation, based on the Richardson solution for the reduced BCS Hamiltonian and applicable over a wide range of the grain sizes and interaction strengths at arbitrary distributions of single-electron energy levels in a grain. Our study is focused upon ultra-small grains, where both the mean value of the nearest-neighbor spacing of single-electron energy levels in a grain and variations of this spacing from grain to grain significantly exceed the superconducting gap in bulk samples of the same material. For these ultra-small superconductor grains, the overall profiles of the magnetic susceptibility as a function of magnetic field and temperature are demonstrated to be qualitatively different from those for normal grains. We show that the analyzed signatures of pairing correlations are sufficiently stable with respect to variations of the average value of the grain size and its dispersion over an assembly of nanograins. The presence of these signatures does not depend on a particular choice of statistics, obeyed by single-electron energy levels in grains.Comment: 40 pages, 12 figures, submitted to Phys. Rev. B, E-mail addresses: [email protected], [email protected], [email protected]

Causality vs. Ward identity in disordered electron systems

We address the problem of fulfilling consistency conditions in solutions for disordered noninteracting electrons. We prove that if we assume the existence of the diffusion pole in an electron-hole symmetric theory we cannot achieve a solution with a causal self-energy that would fully fit the Ward identity. Since the self-energy must be causal, we conclude that the Ward identity is partly violated in the diffusive transport regime of disordered electrons. We explain this violation in physical terms and discuss its consequences.Comment: 4 pages, REVTeX, 6 EPS figure

Minimizing Higgs Potentials via Numerical Polynomial Homotopy Continuation

The study of models with extended Higgs sectors requires to minimize the corresponding Higgs potentials, which is in general very difficult. Here, we apply a recently developed method, called numerical polynomial homotopy continuation (NPHC), which guarantees to find all the stationary points of the Higgs potentials with polynomial-like nonlinearity. The detection of all stationary points reveals the structure of the potential with maxima, metastable minima, saddle points besides the global minimum. We apply the NPHC method to the most general Higgs potential having two complex Higgs-boson doublets and up to five real Higgs-boson singlets. Moreover the method is applicable to even more involved potentials. Hence the NPHC method allows to go far beyond the limits of the Gr\"obner basis approach.Comment: 9 pages, 4 figure

2*2 random matrix ensembles with reduced symmetry: From Hermitian to PT-symmetric matrices

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity-time- (PT-) symmetric matrices. To illustrate the main idea, we first study 2*2 complex Hermitian matrix ensembles with O(2) invariant constraints, yielding novel level-spacing statistics such as singular distributions, half-Gaussian distribution, distributions interpolating between GOE (Gaussian Orthogonal Ensemble) distribution and half Gaussian distributions, as well as gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2*2 PT-symmetric matrix ensembles with real eigenvalues. In particular, PT-symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of PT-symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian Unitary Ensemble) statistics or "truncated-GUE" statistics

Competition and cooperation:aspects of dynamics in sandpiles

In this article, we review some of our approaches to granular dynamics, now well known to consist of both fast and slow relaxational processes. In the first case, grains typically compete with each other, while in the second, they cooperate. A typical result of {\it cooperation} is the formation of stable bridges, signatures of spatiotemporal inhomogeneities; we review their geometrical characteristics and compare theoretical results with those of independent simulations. {\it Cooperative} excitations due to local density fluctuations are also responsible for relaxation at the angle of repose; the {\it competition} between these fluctuations and external driving forces, can, on the other hand, result in a (rare) collapse of the sandpile to the horizontal. Both these features are present in a theory reviewed here. An arena where the effects of cooperation versus competition are felt most keenly is granular compaction; we review here a random graph model, where three-spin interactions are used to model compaction under tapping. The compaction curve shows distinct regions where 'fast' and 'slow' dynamics apply, separated by what we have called the {\it single-particle relaxation threshold}. In the final section of this paper, we explore the effect of shape -- jagged vs. regular -- on the compaction of packings near their jamming limit. One of our major results is an entropic landscape that, while microscopically rough, manifests {\it Edwards' flatness} at a macroscopic level. Another major result is that of surface intermittency under low-intensity shaking.Comment: 36 pages, 23 figures, minor correction

Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution

We exploit a relation between the mean number $N_{m}$ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $\mu$ through the critical value $\mu_c$ we analyse in detail how $N_{m}(\mu)$ drops from being exponentially large in the glassy phase to $N_{m}(\mu)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(\mu)$ in both glassy and simple phases. The width $\delta{\mu}/\mu_c$ of the critical region is found to scale as $N^{-1/3}$ and inside that region $N_{m}(\mu)$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution

Chern-Simons theory, exactly solvable models and free fermions at finite temperature

We show that matrix models in Chern-Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian matrix model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wavefunctions and ground-state energies and point out that the models can be interpreted as quasi-1D Coulomb plasmas. We also study the relationship between Chern-Simons theory on $S^3$ and a system of N one-dimensional fermions at finite temperature with harmonic confinement. In particular we show that the Chern-Simons partition function can be described by the density matrix of the free fermions in a very particular, crystalline, configuration. For this, we both use the Brownian motion and the matrix model description of Chern-Simons theory and find several common features with c=1 theory at finite temperature. Finally, using the exactly solvable model result, we show that the finite temperature effect can be described with a specific two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at large separations.Comment: 19 pages, v2: references adde