98,720 research outputs found
Wilsonian renormalization, differential equations and Hopf algebras
In this paper, we present an algebraic formalism inspired by Butcher's
B-series in numerical analysis and the Connes-Kreimer approach to perturbative
renormalization. We first define power series of non linear operators and
propose several applications, among which the perturbative solution of a fixed
point equation using the non linear geometric series. Then, following
Polchinski, we show how perturbative renormalization works for a non linear
perturbation of a linear differential equation that governs the flow of
effective actions. Then, we define a general Hopf algebra of Feynman diagrams
adapted to iterations of background field effective action computations. As a
simple combinatorial illustration, we show how these techniques can be used to
recover the universality of the Tutte polynomial and its relation to the
-state Potts model. As a more sophisticated example, we use ordered diagrams
with decorations and external structures to solve the Polchinski's exact
renormalization group equation. Finally, we work out an analogous construction
for the Schwinger-Dyson equations, which yields a bijection between planar
diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given
at the conference "Combinatorics and physics" held at Max Planck Institut
fuer Mathematik in Bonn in march 2007, some misprints correcte
Effective action and density functional theory
The effective action for the charge density and the photon field is proposed
as a generalization of the density functional. A simple definition is given for
the density functional, as the functional Legendre transform of the generator
functional of connected Green functions for the density and the photon field,
offering systematic approximation schemes. The leading order of the
perturbation expansion reproduces the Hartree-Fock equation. A renormalization
group motivated method is introduced to turn on the Coulomb interaction
gradually and to find corrections to the Hartree-Fock and the Kohn-Sham
schemes.Comment: New references and a numerical algorithm added, to appear in Phys.
Rev. B. 30 pages, no figure
Transfer Theorems and Asymptotic Distributional Results for m-ary Search Trees
We derive asymptotics of moments and identify limiting distributions, under
the random permutation model on m-ary search trees, for functionals that
satisfy recurrence relations of a simple additive form. Many important
functionals including the space requirement, internal path length, and the
so-called shape functional fall under this framework. The approach is based on
establishing transfer theorems that link the order of growth of the input into
a particular (deterministic) recurrence to the order of growth of the output.
The transfer theorems are used in conjunction with the method of moments to
establish limit laws. It is shown that (i) for small toll sequences
[roughly, ] we have asymptotic normality if and
typically periodic behavior if ; (ii) for moderate toll sequences
[roughly, but ] we have convergence to
non-normal distributions if (where ) and typically
periodic behavior if ; and (iii) for large toll sequences
[roughly, ] we have convergence to non-normal distributions
for all values of m.Comment: 35 pages, 1 figure. Version 2 consists of expansion and rearragement
of the introductory material to aid exposition and the shortening of
Appendices A and B.
Trees, functional equations, and combinatorial Hopf algebras
One of the main virtues of trees is to represent formal solutions of various
functional equations which can be cast in the form of fixed point problems.
Basic examples include differential equations and functional (Lagrange)
inversion in power series rings. When analyzed in terms of combinatorial Hopf
algebras, the simplest examples yield interesting algebraic identities or
enumerative results.Comment: 14 pages, LaTE
Differential Renormalization-Group Approach to the Layered sine-Gordon Model
New qualitative picture of vortex length-scale dependence has been found in
recent electrical transport measurements performed on strongly anisotropic
BSCCO single crystals in zero magnetic field. This indicates the need for a
better description of the 3D/2D crossover in vortex dimensionality. The
vortex-dominated properties of high transition temperature superconductors with
extremely high anisotropy (layered systems) are reasonably well described in
the framework of the layered XY model which can be mapped onto the layered
sine-Gordon model. For the latter we derive an exact renormalization group (RG)
equation using Wegner's and Houghton's approach in the local potential
approximation. The agreement of the UV scaling laws find by us by linearizing
the RG equations with those obtained previously in the literature in the dilute
gas approximation makes the improvement appearant which can be achieved by
solving our RG equations numerically.Comment: 12 pages, no figures, to be published in Philos. Ma
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