On dynamics of volatilities in nonstationary GARCH models

This paper precisely characterizes asymptotic behaviors of the volatilities in nonstationary GARCH(1, 1) models. After suitable renormalization, it is shown that the volatility converges in distribution to a non-degenerate limit. This provides more insight into the dynamics of volatilities in nonstationary GARCH models.     

Multiscale representation of generating and correlation functions for some models of statistical mechanics and quantum field theory

We consider models of statistical mechanics and quantum field theory (in the Euclidean formulation) which are treated using renormalization group methods and where the action is a small perturbation of a quadratic action. We obtain multiscale formulas for the generating and correlation functions aftern renormalization group transformations which bring out the relation with thenth effective action. We derive and compare the formulas for different RGs. The formulas for correlation functions involve (1) two propagators which are determined by a sequence of approximate wave function renormalization constants and renormalization group operators associated with the decomposition into scales of the quadratic form and (2) field derivatives of the nth effective action. For the case of the block field -function RG the formulas are especially simple and for asymptotic free theories only the derivatives at zero field are needed; the formulas have been previously used directly to obtain bounds on correlation functions using information obtained from the analysis of effective actions. The simplicity can be traced to an orthogonality-of-scales property which follows from an implicit wavelet structure. Other commonly used RGs do not have the orthogonality of scales property.     

Generalized renormalization group treatment of the growth kinetics of unstable systems

We discuss the generalization of a renormalization group technique developed previously for the study of ordering in unstable systems in the context of the ferromagnetic Ising model with spin-flip dynamics. Difficulties encountered in earlier work are eliminated through the use of new recursion relations dependent on a continuous spatial rescaling factorb 1. A more careful analysis and implementation of the approximation scheme is carried out. Our improved method allows the study of the anisotropy of the time-dependent structure factor and the pre-scaling behavior of the shape function.     

Renormalization group evaluation of exponents in family name distributions

According to many phenomenological and theoretical studies the distribution of family name frequencies in a population can be asymptotically described by a power law. We show that the Galton-Watson process corresponding to the dynamics of a growing population can be represented in Hilbert space, and its time evolution may be analyzed by renormalization group techniques, thus explaining the origin of the power law and establishing the connection between its exponent and the ratio between the population growth and the name production rates.     

Length-scale dependent superconductor-insulator quantum phase transitions in one dimension

We study the dissipation physics of a one dimensional mesoscopic superconducting quantum interference device array using the field-theoretical renormalization group method. We observe length scale dependent, superconductor-insulator quantum phase transition at very low temperature and also observe the dual behavior of the system for higher and lower values of magnetic field. At a critical magnetic field, we also observe a critical behavior where the resistance is independent of length.     

Chiral perturbation theory: Introduction and recent results in the one-nucleon sector

We provide an introduction to the basic concepts of chiral perturbation theory and discuss some recent developments in the manifestly Lorentz-invariant formulation of the one-nucleon sector.     

Universal features of polymer shapes in crowded environments

We study the universal characteristics of the shape of a polymer chain in an environment with correlated structural obstacles, applying the field-theoretical renormalization group approach. Our results qualitatively indicate an increase of the asymmetry of the polymer shape in crowded environment comparing with the pure solution case.     

Short distance freedom of quantum gravity

Fourth order derivative gravity in 3+1$3+1$ dimensions is perturbatively renormalizable and is shown to describe a unitary theory of gravitons in a limited coupling parameter space. The running gravitational constant which includes graviton contribution is computed. Generically, gravitational Newton?s constant vanishes at short distances in this perturbatively renormalizable and unitary theory.     

Renormalizability conditions for almost-commutative geometries

We formulate conditions for almost-commutative (spacetime) manifolds under which the asymptotically expanded spectral action is renormalizable. These conditions are of a graph-theoretical nature, involving the Krajewski diagrams that classify such geometries. This applies in particular to the Standard Model of particle physics, giving a graph-theoretical argument for its renormalizability. A promising potential application is in the selection of physical (renormalizable) field theories described by almost-commutative geometries, thereby going beyond the Standard Model.