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 $\mathcal{N}$ of functionals of the fields. In this paper, we develop a general method---localisation---to approximate an element of $\mathcal{N}$ by a local polynomial in the fields. From the point of view of the renormalisation group, the construction of the local polynomial corresponding to $F$ in $\mathcal{N}$ amounts to the extraction of the relevant and marginal parts of $F$. We prove estimates relating $F$ and its corresponding local polynomial, in terms of the $T_{\phi}$ 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 $\mathbb{Z}^d$. 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 $d = 4$. 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