1,470 research outputs found
Trees, forests and jungles: a botanical garden for cluster expansions
Combinatoric formulas for cluster expansions have been improved many times
over the years. Here we develop some new combinatoric proofs and extensions of
the tree formulas of Brydges and Kennedy, and test them on a series of
pedagogical examples.Comment: 37 pages, Ecole Polytechnique A-325.099
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
A renormalisation group method. II. Approximation by local polynomials
This paper is the second in a series devoted to the development of a rigorous
renormalisation group method for lattice field theories involving boson fields,
fermion fields, or both. The method is set within a normed algebra
of functionals of the fields. In this paper, we develop a general
method---localisation---to approximate an element of by a local
polynomial in the fields. From the point of view of the renormalisation group,
the construction of the local polynomial corresponding to in
amounts to the extraction of the relevant and marginal parts of . We prove
estimates relating and its corresponding local polynomial, in terms of the
semi-norm introduced in part I of the series.Comment: 30 page
A simple method for finite range decomposition of quadratic forms and Gaussian fields
We present a simple method to decompose the Green forms corresponding to a
large class of interesting symmetric Dirichlet forms into integrals over
symmetric positive semi-definite and finite range (properly supported) forms
that are smoother than the original Green form. This result gives rise to
multiscale decompositions of the associated Gaussian free fields into sums of
independent smoother Gaussian fields with spatially localized correlations. Our
method makes use of the finite propagation speed of the wave equation and
Chebyshev polynomials. It improves several existing results and also gives
simpler proofs.Comment: minor correction for t<
A renormalisation group method. IV. Stability analysis
This paper is the fourth in a series devoted to the development of a rigorous
renormalisation group method for lattice field theories involving boson fields,
fermion fields, or both. The third paper in the series presents a perturbative
analysis of a supersymmetric field theory which represents the continuous-time
weakly self-avoiding walk on . We now present an analysis of the
relevant interaction functional of the supersymmetric field theory, which
permits a nonperturbative analysis to be carried out in the critical dimension
. The results in this paper include: proof of stability of the
interaction, estimates which enable control of Gaussian expectations involving
both boson and fermion fields, estimates which bound the errors in the
perturbative analysis, and a crucial contraction estimate to handle irrelevant
directions in the flow of the renormalisation group. These results are
essential for the analysis of the general renormalisation group step in the
fifth paper in the series.Comment: 62 page
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