The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincar\'e--Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincar\'e--Liapunov method.Comment: 24 pages, no figure

Decoherence, instantons, and cosmological horizons

We consider the possibility that tunneling in a degenerate double-well potential in de Sitter spacetime leads to coherent oscillations of quantum probability to find the system in a given well. We concentrate on the case when the mass scale of the potential is much larger than the Hubble parameter and present a procedure for analytical continuation of the ``time''-dependent instantons, which allows us to study the subsequent real-time evolution. We find that the presence of the de Sitter horizon makes tunneling completely incoherent and calculate the decoherence time. We discuss the difference between this case and the case of a $\theta$-vacuum, when tunneling in de Sitter spacetime preserves quantum coherence.Comment: 21 pages, latex, 3 figure

New attractor mechanism for spherically symmetric extremal black holes

We introduce a new attractor mechanism to find the entropy for spherically symmetric extremal black holes. The key ingredient is to find a two-dimensional (2D) dilaton gravity with the dilaton potential $V(\phi)$. The condition of an attractor is given by $\nabla^2\phi=V(\phi_0)$ and $\bar{R}_2=-V^{\prime}(\phi_0)$ and for a constant dilaton $\phi=\phi_0$, these are also used to find the location of the degenerate horizon $r=r_{e}$ of an extremal black hole. As a nontrivial example, we consider an extremal regular black hole obtained from the coupled system of Einstein gravity and nonlinear electrodynamics. The desired Bekenstein-Hawking entropy is successfully recovered from the generalized entropy formula combined with the 2D dilaton gravity, while the entropy function approach does not work for obtaining this entropy.Comment: 20 pages, 4 figures, Accepted for publication in Physical Review D. This version includes revisions suggested by the refere

Complete Wetting of Gluons and Gluinos

Complete wetting is a universal phenomenon associated with interfaces separating coexisting phases. For example, in the pure gluon theory, at $T_c$ an interface separating two distinct high-temperature deconfined phases splits into two confined-deconfined interfaces with a complete wetting layer of confined phase between them. In supersymmetric Yang-Mills theory, distinct confined phases may coexist with a Coulomb phase at zero temperature. In that case, the Coulomb phase may completely wet a confined-confined interface. Finally, at the high-temperature phase transition of gluons and gluinos, confined-confined interfaces are completely wet by the deconfined phase, and similarly, deconfined-deconfined interfaces are completely wet by the confined phase. For these various cases, we determine the interface profiles and the corresponding complete wetting critical exponents. The exponents depend on the range of the interface interactions and agree with those of corresponding condensed matter systems.Comment: 15 pages, 5 figure

Geometrical Ambiguity of Pair Statistics. I. Point Configurations

Point configurations have been widely used as model systems in condensed matter physics, materials science and biology. Statistical descriptors such as the $n$-body distribution function $g_n$ is usually employed to characterize the point configurations, among which the most extensively used is the pair distribution function $g_2$. An intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration. Although it is known that the pair-distance information contained in $g_2$ is in general insufficient to uniquely determine a point configuration, this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies. In this paper, we introduce the idea of the distance space, called the $\mathbb{D}$ space. The pair distances of a specific point configuration are then represented by a single point in the $\mathbb{D}$ space. We derive the conditions on the pair distances that can be associated with a point configuration, which are equivalent to the realizability conditions of the pair distribution function $g_2$. Moreover, we derive the conditions on the pair distances that can be assembled into distinct configurations. These conditions define a bounded region in the $\mathbb{D}$ space. By explicitly constructing a variety of degenerate point configurations using the $\mathbb{D}$ space, we show that pair information is indeed insufficient to uniquely determine the configuration in general. We also discuss several important problems in statistical physics based on the $\mathbb{D}$ space.Comment: 28 pages, 8 figure

How to escape Aharonov-Bohm cages ?

We study the effect of disorder and interactions on a recently proposed magnetic field induced localization mechanism. We show that both partially destroy the extreme confinement of the excitations occuring in the pure case and give rise to unusual behavior. We also point out the role of the edge states that allows for a propagation of the electrons in these systems.Comment: 22 pages, 20 EPS figure