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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 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
The strong interaction limit of continuous-time weakly self-avoiding walk
The strong interaction limit of the discrete-time weakly self-avoiding walk
(or Domb--Joyce model) is trivially seen to be the usual strictly self-avoiding
walk. For the continuous-time weakly self-avoiding walk, the situation is more
delicate, and is clarified in this paper. The strong interaction limit in the
continuous-time setting depends on how the fugacity is scaled, and in one
extreme leads to the strictly self-avoiding walk, in another to simple random
walk. These two extremes are interpolated by a new model of a self-repelling
walk that we call the "quick step" model. We study the limit both for walks
taking a fixed number of steps, and for the two-point function
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