Peripheral fillings of relatively hyperbolic groups

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $\pi_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Various applications of these results are discussed.Comment: The difference with the previous version is that Proposition 3.2 is proved for quasi--geodesics instead of geodesics. This allows to simplify the exposition in the last section. To appear in Invent. Mat

First passage time exponent for higher-order random walks:Using Levy flights

We present a heuristic derivation of the first passage time exponent for the integral of a random walk [Y. G. Sinai, Theor. Math. Phys. {\bf 90}, 219 (1992)]. Building on this derivation, we construct an estimation scheme to understand the first passage time exponent for the integral of the integral of a random walk, which is numerically observed to be $0.220\pm0.001$. We discuss the implications of this estimation scheme for the $n{\rm th}$ integral of a random walk. For completeness, we also address the $n=\infty$ case. Finally, we explore an application of these processes to an extended, elastic object being pulled through a random potential by a uniform applied force. In so doing, we demonstrate a time reparameterization freedom in the Langevin equation that maps nonlinear stochastic processes into linear ones.Comment: 4 figures, submitted to PR

Progressive collapse of multi-storey buildings due to sudden column loss—Part II: Application

Accepted versio

Renormalization Group Evolution of Dirac Neutrino Masses

There are good reasons why neutrinos could be Majorana particles, but there exist also a number of very good reasons why neutrinos could have Dirac masses. The latter option deserves more attention and we derive therefore analytic expressions describing the renormalization group evolution of mixing angles and of the CP phase for Dirac neutrinos. Radiative corrections to leptonic mixings are in this case enhanced compared to the quark mixings because the hierarchy of neutrino masses is milder and because the mixing angles are larger. The renormalization group effects are compared to the precision of current and future neutrino experiments. We find that, in the MSSM framework, radiative corrections of the mixing angles are for large \tan\beta comparable to the precision of future experiments.Comment: 19 pages, 5 figures; error in eq. 8 corrected, references adde

Theta-13 as a Probe of Mu-Tau symmetry for Leptons

Many experiments are being planned to measure the neutrino mixing parameter $\theta_{13}$ using reactor as well as accelerator neutrino beams. In this note, the theoretical significance of a high precision measurement of this parameter is discussed. It is emphasized that it will provide crucial information about different ways to understand the origin of large atmospheric neutrino mixing and move us closer towards determining the neutrino mass matrix. For instance if exact $\mu\leftrightarrow \tau$ symmetry in the neutrino mass matrix is assumed to be the reason for maximal $\nu_\mu-\nu_\tau$ mixing, one gets $\theta_{13}=0$. Whether $\theta_{13}\simeq \sqrt{\Delta m^2_{\odot}/\Delta m^2_A}$ or $\theta_{13}\simeq \Delta m^2_{\odot}/\Delta m^2_A$ can provide information about the way the $\mu\leftrightarrow \tau$ symmetry breaking manifests in the case of normal hierarchy. We also discuss the same question for inverted hierarchy as well as possible gauge theories with this symmetry.Comment: 12 pages; no figures; latex; more exact expressions given for some parameters and minor typos corrected; paper accepted for publication in JHE

Views of the Chiral Magnetic Effect

My personal views of the Chiral Magnetic Effect are presented, which starts with a story about how we came up with the electric-current formula and continues to unsettled subtleties in the formula. There are desirable features in the formula of the Chiral Magnetic Effect but some considerations would lead us to even more questions than elucidations. The interpretation of the produced current is indeed very non-trivial and it involves a lot of confusions that have not been resolved.Comment: 19 pages, no figure; typos corrected, references significantly updated, to appear in Lect. Notes Phys. "Strongly interacting matter in magnetic fields" (Springer), edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Ye

Natural Orbitals and BEC in traps, a diffusion Monte Carlo analysis

We investigate the properties of hard core Bosons in harmonic traps over a wide range of densities. Bose-Einstein condensation is formulated using the one-body Density Matrix (OBDM) which is equally valid at low and high densities. The OBDM is calculated using diffusion Monte Carlo methods and it is diagonalized to obtain the "natural" single particle orbitals and their occupation, including the condensate fraction. At low Boson density, $na^3 < 10^{-5}$, where $n = N/V$ and $a$ is the hard core diameter, the condensate is localized at the center of the trap. As $na^3$ increases, the condensate moves to the edges of the trap. At high density it is localized at the edges of the trap. At $na^3 \leq 10^{-4}$ the Gross-Pitaevskii theory of the condensate describes the whole system within 1%. At $na^3 \approx 10^{-3}$ corrections are 3% to the GP energy but 30% to the Bogoliubov prediction of the condensate depletion. At $na^3 \gtrsim 10^{-2}$, mean field theory fails. At $na^3 \gtrsim 0.1$, the Bosons behave more like a liquid $^4$He droplet than a trapped Boson gas.Comment: 13 pages, 14 figures, submitted Phys. Rev.

Resonant structure of space-time of early universe

A new fully quantum method describing penetration of packet from internal well outside with its tunneling through the barrier of arbitrary shape used in problems of quantum cosmology, is presented. The method allows to determine amplitudes of wave function, penetrability $T_{\rm bar}$ and reflection $R_{\rm bar}$ relatively the barrier (accuracy of the method: $|T_{\rm bar}+R_{\rm bar}-1| < 1 \cdot 10^{-15}$), coefficient of penetration (i.e. probability of the packet to penetrate from the internal well outside with its tunneling), coefficient of oscillations (describing oscillating behavior of the packet inside the internal well). Using the method, evolution of universe in the closed Friedmann--Robertson--Walker model with quantization in presence of positive cosmological constant, radiation and component of generalize Chaplygin gas is studied. It is established (for the first time): (1) oscillating dependence of the penetrability on localization of start of the packet; (2) presence of resonant values of energy of radiation $E_{\rm rad}$, at which the coefficient of penetration increases strongly. From analysis of these results it follows: (1) necessity to introduce initial condition into both non-stationary, and stationary quantum models; (2) presence of some definite values for the scale factor $a$, where start of expansion of universe is the most probable; (3) during expansion of universe in the initial stage its radius is changed not continuously, but passes consequently through definite discrete values and tends to continuous spectrum in latter time.Comment: 18 pages, 14 figures, 4 table

Green function techniques in the treatment of quantum transport at the molecular scale

The theoretical investigation of charge (and spin) transport at nanometer length scales requires the use of advanced and powerful techniques able to deal with the dynamical properties of the relevant physical systems, to explicitly include out-of-equilibrium situations typical for electrical/heat transport as well as to take into account interaction effects in a systematic way. Equilibrium Green function techniques and their extension to non-equilibrium situations via the Keldysh formalism build one of the pillars of current state-of-the-art approaches to quantum transport which have been implemented in both model Hamiltonian formulations and first-principle methodologies. We offer a tutorial overview of the applications of Green functions to deal with some fundamental aspects of charge transport at the nanoscale, mainly focusing on applications to model Hamiltonian formulations.Comment: Tutorial review, LaTeX, 129 pages, 41 figures, 300 references, submitted to Springer series "Lecture Notes in Physics