Network synchronization: Optimal and Pessimal Scale-Free Topologies

By employing a recently introduced optimization algorithm we explicitely design optimally synchronizable (unweighted) networks for any given scale-free degree distribution. We explore how the optimization process affects degree-degree correlations and observe a generic tendency towards disassortativity. Still, we show that there is not a one-to-one correspondence between synchronizability and disassortativity. On the other hand, we study the nature of optimally un-synchronizable networks, that is, networks whose topology minimizes the range of stability of the synchronous state. The resulting ``pessimal networks'' turn out to have a highly assortative string-like structure. We also derive a rigorous lower bound for the Laplacian eigenvalue ratio controlling synchronizability, which helps understanding the impact of degree correlations on network synchronizability.Comment: 11 pages, 4 figs, submitted to J. Phys. A (proceedings of Complex Networks 2007

Thiemann transform for gravity with matter fields

The generalised Wick transform discovered by Thiemann provides a well-established relation between the Euclidean and Lorentzian theories of general relativity. We extend this Thiemann transform to the Ashtekar formulation for gravity coupled with spin-1/2 fermions, a non-Abelian Yang-Mills field, and a scalar field. It is proved that, on functions of the gravitational and matter phase space variables, the Thiemann transform is equivalent to the composition of an inverse Wick rotation and a constant complex scale transformation of all fields. This result holds as well for functions that depend on the shift vector, the lapse function, and the Lagrange multipliers of the Yang-Mills and gravitational Gauss constraints, provided that the Wick rotation is implemented by means of an analytic continuation of the lapse. In this way, the Thiemann transform is furnished with a geometric interpretation. Finally, we confirm the expectation that the generator of the Thiemann transform can be determined just from the spin of the fields and give a simple explanation for this fact.Comment: LaTeX 2.09, 14 pages, no figure

Dynamics of cosmic strings and springs; a covariant formulation

A general family of charge-current carrying cosmic string models is investigated. In the special case of circular configurations in arbitrary axially symmetric gravitational and electromagnetic backgrounds the dynamics is determined by simple point particle Hamiltonians. A certain "duality" transformation relates our results to previous ones, obtained by Carter et. al., for an infinitely long open stationary string in an arbitrary stationary background.Comment: 11 pages, Latex, Nordita preprint 93/28

Valence transition in the periodic Anderson model

A very rich phase diagram has recently been found in CeCu$_{2}$Si$_{2}$ from high pressure experiments where, in particular, a transition between an intermediate valence configuration and an integral valent heavy fermion state has been observed. We show that such a valence transition can be understood in the framework of the periodic Anderson model. In particular, our results show a breakdown of a mixed-valence state which is accompanied by a drastic change in the \textit{f} occupation in agreement with experiment. This valence transition can possibly be interpreted as a collapse of the large Fermi surface of the heavy fermion state which incorporates not only the conduction electrons but also the localized \textit{f} electrons. The theoretical approach used in this paper is based on the novel projector-based renormalization method (PRM). With respect to the periodic Anderson model, the method was before only employed in combination with the basic approximations of the well-known slave-boson mean-field theory. In this paper, the PRM treatment is performed in a more sophisticated manner where both mixed as well as integral valent solutions have been obtained. Furthermore, we argue that the presented PRM approach might be a promising starting point to study the competing interactions in CeCu$_{2}$Si$_{2}$ and related compounds.Comment: 9 pages, 3 figures included; v2: completely revised and extended versio

Langevin dynamics of the Lebowitz-Percus model

We revisit the hard-spheres lattice gas model in the spherical approximation proposed by Lebowitz and Percus (J. L. Lebowitz, J. K. Percus, Phys. Rev.{\ 144} (1966) 251). Although no disorder is present in the model, we find that the short-range dynamical restrictions in the model induce glassy behavior. We examine the off-equilibrium Langevin dynamics of this model and study the relaxation of the density as well as the correlation, response and overlap two-time functions. We find that the relaxation proceeds in two steps as well as absence of anomaly in the response function. By studying the violation of the fluctuation-dissipation ratio we conclude that the glassy scenario of this model corresponds to the dynamics of domain growth in phase ordering kinetics.Comment: 21 pages, RevTeX, 14 PS figure

Microscopic Black Hole Pairs in Highly-Excited States

We consider the quantum mechanics of a system consisting of two identical, Planck-size Schwarzschild black holes revolving around their common center of mass. We find that even in a very highly-excited state such a system has very sharp, discrete energy eigenstates, and the system performs very rapid transitions from a one stationary state to another. For instance, when the system is in the 100th excited state, the life times of the energy eigenstates are of the order of $10^{-30}$ s, and the energies of gravitons released in transitions between nearby states are of the order of $10^{22}$ eV.Comment: 22 pages, 3 figures, uses RevTe

Causality violation and singularities

We show that singularities necessarily occur when a boundary of causality violating set exists in a space-time under the physically suitable assumptions except the global causality condition in the Hawking-Penrose singularity theorems. Instead of the global causality condition, we impose some restrictions on the causality violating sets to show the occurrence of singularities.Comment: 11 pages, latex, 2 eps figure

S3 x Z2 model for neutrino mass matrices

We propose a model for lepton mass matrices based on the seesaw mechanism, a complex scalar gauge singlet and a horizontal symmetry S_3 \times \mathbbm{Z}_2. In a suitable weak basis, the charged-lepton mass matrix and the neutrino Dirac mass matrix are diagonal, but the vacuum expectation value of the scalar gauge singlet renders the Majorana mass matrix of the right-handed neutrinos non-diagonal, thereby generating lepton mixing. When the symmetry $S_3$ is not broken in the scalar potential, the effective light-neutrino Majorana mass matrix enjoys $\mu$--$\tau$ interchange symmetry, thus predicting maximal atmospheric neutrino mixing together with $U_{e3} = 0$. A partial and less predictive form of $\mu$--$\tau$ interchange symmetry is obtained when the symmetry $S_3$ is softly broken in the scalar potential. Enlarging the symmetry group S_3 \times \mathbbm{Z}_2 by an additional discrete electron-number symmetry \mathbbm{Z}_2^{(e)}, a more predicitive model is obtained, which is in practice indistinguishable from a previous one based on the group $D_4$.Comment: 13 pages, 3 figures, final version for publication in JHE

Critical Exponents of the Three Dimensional Random Field Ising Model

The phase transition of the three--dimensional random field Ising model with a discrete ($\pm h$) field distribution is investigated by extensive Monte Carlo simulations. Values of the critical exponents for the correlation length, specific heat, susceptibility, disconnected susceptibility and magnetization are determined simultaneously via finite size scaling. While the exponents for the magnetization and disconnected susceptibility are consistent with a first order transition, the specific heat appears to saturate indicating no latent heat. Sample to sample fluctuations of the susceptibilty are consistent with the droplet picture for the transition.Comment: Revtex, 10 pages + 4 figures included as Latex files and 1 in Postscrip