414 research outputs found
The Exact Renormalization Group
This is a very brief introduction to Wilson's Renormalization Group with
emphasis on mathematical developments.Comment: 17 pages, AMS LaTeX. Contribution to the Encyclopedia of Mathematical
Physics (Elsevier, 2006). Typos, journal reference correcte
Renormalization group approach to interacting polymerised manifolds
We propose to study the infrared behaviour of polymerised (or tethered)
random manifolds of dimension D interacting via an exclusion condition with a
fixed impurity in d-dimensional Euclidean space in which the manifold is
embedded. We prove rigorously, via methods of Wilson's renormalization group,
the convergence to a non Gaussian fixed point for suitably chosen physical
parameters.Comment: 90 pages, Plain tex file. Updated version with more detailed
introduction and added reference
Finite range Decomposition of Gaussian Processes
Let \D be the finite difference Laplacian associated to the lattice
\bZ^{d}. For dimension , and a sufficiently large
positive dyadic integer, we prove that the integral kernel of the resolvent
G^{a}:=(a-\D)^{-1} can be decomposed as an infinite sum of positive
semi-definite functions of finite range, for
. Equivalently, the Gaussian process on the lattice with
covariance admits a decomposition into independent Gaussian processes
with finite range covariances. For , has a limiting scaling form
as .
As a corollary, such decompositions also exist for fractional powers
(-\D)^{-\alpha/2}, . The results of this paper give an
alternative to the block spin renormalization group on the lattice.Comment: 26 pages, LaTeX, paper in honour of G.Jona-Lasinio.Typos corrected,
corrections in section 5 and appendix
CRITICAL (Phi^{4}_{3,\epsilon})
The Euclidean (\phi^{4})_{3,\epsilon model in corresponds to a
perturbation by a interaction of a Gaussian measure on scalar fields
with a covariance depending on a real parameter in the range . For one recovers the covariance of a massless
scalar field in . For is a marginal interaction.
For the covariance continues to be Osterwalder-Schrader and
pointwise positive. After introducing cutoffs we prove that for ,
sufficiently small, there exists a non-gaussian fixed point (with one unstable
direction) of the Renormalization Group iterations. These iterations converge
to the fixed point on its stable (critical) manifold which is constructed.Comment: 49 pages, plain tex, macros include
On the Convergence to the Continuum of Finite Range Lattice Covariances
In J. Stat. Phys. 115, 415-449 (2004) Brydges, Guadagni and Mitter proved the
existence of multiscale expansions of a class of lattice Green's functions as
sums of positive definite finite range functions (called fluctuation
covariances). The lattice Green's functions in the class considered are
integral kernels of inverses of second order positive self adjoint operators
with constant coefficients and fractional powers thereof. The fluctuation
coefficients satisfy uniform bounds and the sequence converges in appropriate
norms to a smooth, positive definite, finite range continuum function. In this
note we prove that the convergence is actually exponentially fast.Comment: 14 pages. We have added further references as well as a proof of
Corollary 2.2. This version submitted for publicatio
On an Information and Control Architecture for Future Electric Energy Systems
This paper presents considerations towards an information and control
architecture for future electric energy systems driven by massive changes
resulting from the societal goals of decarbonization and electrification. This
paper describes the new requirements and challenges of an extended information
and control architecture that need to be addressed for continued reliable
delivery of electricity. It identifies several new actionable information and
control loops, along with their spatial and temporal scales of operation, which
can together meet the needs of future grids and enable deep decarbonization of
the electricity sector. The present architecture of electric power grids
designed in a different era is thereby extensible to allow the incorporation of
increased renewables and other emerging electric loads.Comment: This paper is accepted, to appear in the Proceedings of the IEE
Renormalization of the Hamiltonian and a geometric interpretation of asymptotic freedom
Using a novel approach to renormalization in the Hamiltonian formalism, we
study the connection between asymptotic freedom and the renormalization group
flow of the configuration space metric. It is argued that in asymptotically
free theories the effective distance between configuration decreases as high
momentum modes are integrated out.Comment: 22 pages, LaTeX, no figures; final version accepted in Phys.Rev.D;
added reference and appendix with comment on solution of eq. (9) in the tex
Completeness of Wilson loop functionals on the moduli space of and -connections
The structure of the moduli spaces \M := \A/\G of (all, not just flat)
and connections on a n-manifold is analysed. For any
topology on the corresponding spaces \A of all connections which satisfies
the weak requirement of compatibility with the affine structure of \A, the
moduli space \M is shown to be non-Hausdorff. It is then shown that the
Wilson loop functionals --i.e., the traces of holonomies of connections around
closed loops-- are complete in the sense that they suffice to separate all
separable points of \M. The methods are general enough to allow the
underlying n-manifold to be topologically non-trivial and for connections to be
defined on non-trivial bundles. The results have implications for canonical
quantum general relativity in 4 and 3 dimensions.Comment: Plain TeX, 7 pages, SU-GP-93/4-
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
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