  In this paper, we explore the version of Hairer's regularity structures based
on a greedier index set than trees, as introduced by Otto, Sauer, Smith and
Weber. More precisely, we construct and stochastically estimate the
renormalized model avoiding the use of Feynman diagrams but still in a fully
automated, i. e. inductive way. This is carried out for a class of quasi-linear
parabolic PDEs driven by noise in the full singular but renormalizable range.
We assume a spectral gap inequality on the (not necessarily Gaussian) noise
ensemble. The resulting control on the variance of the model naturally
complements its vanishing expectation arising from the BPHZ-choice of
renormalization. We capture the gain in regularity on the level of the
Malliavin derivative of the model by describing it as a modelled distribution.
Symmetry is an important guiding principle and built-in on the level of the
renormalization Ansatz. Our approach is analytic and top-down rather than
combinatorial and bottom-up.
