13 research outputs found
Linked cluster expansions beyond nearest neighbour interactions: convergence and graph classes
We generalize the technique of linked cluster expansions on hypercubic
lattices to actions that couple fields at lattice sites which are not nearest
neighbours. We show that in this case the graphical expansion can be arranged
in such a way that the classes of graphs to be considered are identical to
those of the pure nearest neighbour interaction. The only change then concerns
the computation of lattice imbedding numbers. All the complications that arise
can be reduced to a generalization of the notion of free random walks,
including hopping beyond nearest neighbour. Explicit expressions for
combinatorical numbers of the latter are given. We show that under some general
conditions the linked cluster expansion series have a non-vanishing radius of
convergence.Comment: 20 pages, latex2
Hierarchical renormalization goup fixed points
Hierarchical renormalization group transformations are related to
non-associative algebras. Non-trivial infrared fixed points are shown to be
solutions of polynomial equations. At the example of a scalar model in
dimensions some methods for the solution of these algebraic equations
are presented.Comment: Contribution to Lat94, 27 Sep -- 1 Oct 1994, Bielefeld, 6 pages,
latex, no figure
On Renormalization Group Flows and Polymer Algebras
In this talk methods for a rigorous control of the renormalization group (RG)
flow of field theories are discussed. The RG equations involve the flow of an
infinite number of local partition functions. By the method of exact
beta-function the RG equations are reduced to flow equations of a finite number
of coupling constants. Generating functions of Greens functions are expressed
by polymer activities. Polymer activities are useful for solving the large
volume and large field problem in field theory. The RG flow of the polymer
activities is studied by the introduction of polymer algebras. The definition
of products and recursive functions replaces cluster expansion techniques.
Norms of these products and recursive functions are basic tools and simplify a
RG analysis for field theories. The methods will be discussed at examples of
the -model, the -model and hierarchical scalar field
theory (infrared fixed points).Comment: 32 pages, LaTeX, MS-TPI-94-12, Talk presented at the conference
``Constructive Results in Field Theory, Statistical Mechanics and Condensed
Matter Physics'', 25-27 July 1994, Palaiseau, Franc
Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their -Expansions
Nontrivial fixed points of the hierarchical renormalization group are
computed by numerically solving a system of quadratic equations for the
coupling constants. This approach avoids a fine tuning of relevant parameters.
We study the eigenvalues of the renormalization group transformation,
linearized around the non-trivial fixed points. The numerical results are
compared with -expansion.Comment: LaTex file, 24 pages, 5 figures appended as 1 PostScript file,
preprint MS-TPI-94-