  In this article we provide the complete proof of the result announced in
arXiv:1210.7717 about the construction of scale invariant non-Gaussian
generalized stochastic processes over three dimensional p-adic space. The
construction includes that of the associated squared field and our result shows
this squared field has a dynamically generated anomalous dimension which
rigorously confirms a prediction made more than forty years ago, in an
essentially identical situation, by K. G. Wilson. We also prove a mild form of
universality for the model under consideration. Our main innovation is that our
rigourous renormalization group formalism allows for space dependent couplings.
We derive the relationship between mixed correlations and the dynamical systems
features of our extended renormalization group transformation at a nontrivial
fixed point. The key to our control of the composite field is a partial
linearization theorem which is an infinite-dimensional version of the Koenigs
Theorem in holomorphic dynamics. This is akin to a nonperturbative construction
of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near
the critical surface. Our presentation is essentially self-contained and geared
towards a wider audience. While primarily concerning the areas of probability
and mathematical physics we believe this article will be of interest to
researchers in dynamical systems theory, harmonic analysis and number theory.
It can also be profitably read by graduate students in theoretical physics with
a craving for mathematical precision while struggling to learn the
renormalization group.
