37 research outputs found
The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3
We consider an Euclidean supersymmetric field theory in given by a
supersymmetric perturbation of an underlying massless Gaussian measure
on scalar bosonic and Grassmann fields with covariance the Green's function of
a (stable) L\'evy random walk in . The Green's function depends on the
L\'evy-Khintchine parameter with . For
the interaction is marginal. We prove for
sufficiently small and initial
parameters held in an appropriate domain the existence of a global
renormalization group trajectory uniformly bounded on all renormalization group
scales and therefore on lattices which become arbitrarily fine. At the same
time we establish the existence of the critical (stable) manifold. The
interactions are uniformly bounded away from zero on all scales and therefore
we are constructing a non-Gaussian supersymmetric field theory on all scales.
The interest of this theory comes from the easily established fact that the
Green's function of a (weakly) self-avoiding L\'evy walk in is a second
moment (two point correlation function) of the supersymmetric measure governing
this model. The control of the renormalization group trajectory is a
preparation for the study of the asymptotics of this Green's function. The
rigorous control of the critical renormalization group trajectory is a
preparation for the study of the critical exponents of the (weakly)
self-avoiding L\'evy walk in .Comment: 82 pages, Tex with macros supplied. Revision includes 1. redefinition
of norms involving fermions to ensure uniqueness. 2. change in the definition
of lattice blocks and lattice polymer activities. 3. Some proofs have been
reworked. 4. New lemmas 5.4A, 5.14A, and new Theorem 6.6. 5.Typos
corrected.This is the version to appear in Journal of Statistical Physic
Kosterlitz-Thouless Transition Line for the Two Dimensional Coulomb Gas
With a rigorous renormalization group approach, we study the pressure of the
two dimensional Coulomb Gas along a small piece of the Kosterlitz-Thouless
transition line, i.e. the boundary of the dipole region in the
activity-temperature phase-space.Comment: 61 pages, 2 figure
Long range order for lattice dipoles
We consider a system of classical Heisenberg spins on a cubic lattice in
dimensions three or more, interacting via the dipole-dipole interaction. We
prove that at low enough temperature the system displays orientational long
range order, as expected by spin wave theory. The proof is based on reflection
positivity methods. In particular, we demonstrate a previously unproven
conjecture on the dispersion relation of the spin waves, first proposed by
Froehlich and Spencer, which allows one to apply infrared bounds for estimating
the long distance behavior of the spin-spin correlation functions.Comment: 9 page
On the convergence of cluster expansions for polymer gases
We compare the different convergence criteria available for cluster
expansions of polymer gases subjected to hard-core exclusions, with emphasis on
polymers defined as finite subsets of a countable set (e.g. contour expansions
and more generally high- and low-temperature expansions). In order of
increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a
simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use
of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via
a direct combinatorial handling of the terms of the expansion. We show that for
subset polymers our sharper criterion can be proven both by a suitable
adaptation of Dobrushin inductive argument and by an alternative --in fact,
more elementary-- handling of the Kirkwood-Salzburg equations. In addition we
show that for general abstract polymers this alternative treatment leads to the
same convergence region as the inductive Dobrushin argument and, furthermore,
to a systematic way to improve bounds on correlations
Quark Confinement and Dual Representation in 2+1 Dimensional Pure Yang-Mills Theory
We study the quark confinement problem in 2+1 dimensional pure Yang-Mills
theory using euclidean instanton methods. The instantons are regularized and
dressed Wu-Yang monopoles. The dressing of a monopole is due to the mean field
of the rest of the monopoles. We argue that such configurations are stable to
small perturbations unlike the case of singular, undressed monopoles. Using
exact non-perturbative results for the 3-dim. Coulomb gas, where Debye
screening holds for arbitrarily low temperatures, we show in a self-consistent
way that a mass gap is dynamically generated in the gauge theory. The mass gap
also determines the size of the monopoles. In a sense the pure Yang-Mills
theory generates a dynamical Higgs effect. We also identify the disorder
operator of the model in terms of the Sine-Gordon field of the Coulomb gas.Comment: 26 pages, RevTex, Title changed, a new section added, the discussion
on stability of dressed monopole expanded. Version to appear in Physical
Review
An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer
lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower
bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q).
The upper bound is based on a conjecture claiming that the p monomer-dimer
entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We
compute the first three terms in the formal asymptotic expansion of
(lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching
conjecture is satisfied for (lambda_d)(p).Comment: 15 pages, much more about d=1,2,
Analysis of path integrals at low temperature : Box formula, occupation time and ergodic approximation
We study the low temperature behaviour of path integrals for a simple
one-dimensional model. Starting from the Feynman-Kac formula, we derive a new
functional representation of the density matrix at finite temperature, in terms
of the occupation times of Brownian motions constrained to stay within boxes
with finite sizes. From that representation, we infer a kind of ergodic
approximation, which only involves double ordinary integrals. As shown by its
applications to different confining potentials, the ergodic approximation turns
out to be quite efficient, especially in the low-temperature regime where other
usual approximations fail
Renormalization group analysis of the 2D Hubbard model
Salmhofer [Commun. Math. Phys. 194, 249 (1998)] has recently developed a new
renormalization group method for interacting Fermi systems, where the complete
flow from the bare action of a microscopic model to the effective low-energy
action, as a function of a continuously decreasing infrared cutoff, is given by
a differential flow equation which is local in the flow parameter. We apply
this approach to the repulsive two-dimensional Hubbard model with nearest and
next-nearest neighbor hopping amplitudes. The flow equation for the effective
interaction is evaluated numerically on 1-loop level. The effective
interactions diverge at a finite energy scale which is exponentially small for
small bare interactions. To analyze the nature of the instabilities signalled
by the diverging interactions we extend Salmhofers renormalization group for
the calculation of susceptibilities. We compute the singlet superconducting
susceptibilities for various pairing symmetries and also charge and spin
density susceptibilities. Depending on the choice of the model parameters
(hopping amplitudes, interaction strength and band-filling) we find
commensurate and incommensurate antiferromagnetic instabilities or d-wave
superconductivity as leading instability. We present the resulting phase
diagram in the vicinity of half-filling and also results for the density
dependence of the critical energy scale.Comment: 16 pages, RevTeX, 16 eps figure
Charge and Current Sum Rules in Quantum Media Coupled to Radiation
This paper concerns the equilibrium bulk charge and current density
correlation functions in quantum media, conductors and dielectrics, fully
coupled to the radiation (the retarded regime). A sequence of static and
time-dependent sum rules, which fix the values of certain moments of the charge
and current density correlation functions, is obtained by using Rytov's
fluctuational electrodynamics. A technique is developed to extract the
classical and purely quantum-mechanical parts of these sum rules. The sum rules
are critically tested in the classical limit and on the jellium model. A
comparison is made with microscopic approaches to systems of particles
interacting through Coulomb forces only (the non-retarded regime). In contrast
with microscopic results, the current-current correlation function is found to
be integrable in space, in both classical and quantum regimes.Comment: 19 pages, 1 figur
From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the L\'evy area of fractional Brownian motion with Hurst index
{Let be a -dimensional fractional Brownian motion
with Hurst index , or more generally a Gaussian process whose paths
have the same local regularity. Defining properly iterated integrals of is
a difficult task because of the low H\"older regularity index of its paths. Yet
rough path theory shows it is the key to the construction of a stochastic
calculus with respect to , or to solving differential equations driven by
.
We intend to show in a series of papers how to desingularize iterated
integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure
defined by a limit in law procedure. Convergence is proved by using "standard"
tools of constructive field theory, in particular cluster expansions and
renormalization. These powerful tools allow optimal estimates, and call for an
extension of Gaussian tools such as for instance the Malliavin calculus.
After a first introductory paper \cite{MagUnt1}, this one concentrates on the
details of the constructive proof of convergence for second-order iterated
integrals, also known as L\'evy area