  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.
