Rigorous renormalization group (RG) methods give a powerful approach to problems of statistical mechanics and constructive quantum field theory (QFT), e.g., allowing a successful treatment of critical or just renormalizable models like the massive Gross-Neveu model in two dimensions. However, the proofs in the area tend to be very long and technically demanding. The hierarchical setting, ideal for a first introduction to the RG, provides a simplified toy model where the conceptual picture is cleaner and easier to understand.
It even comes with a natural analogue of the notion of conformal invariance, which may be useful as a testing ground for new ideas and methods for a still elusive RG approach to conformal QFT. In these lectures, I will explain the RG approach to constructing the UV limit in finite volume of the three-dimensional phi-four model, in this simpler hierarchical context. Just as in the Euclidean (non-hierarchical) situation, the rudimentary renormalizations provided by Wick ordering are not enough, and an additional logarithmic correction to the mass or phi-squared term is needed to remove the UV cutoff.
I will begin by introducing the hierarchical setting where the random Schwartz distributions being studied live on the boundary of an infinite regular tree instead of the three-dimensional Euclidean space. I will explain the analogue of the GFF (Gaussian free field) or more generally the FGF (fractional Gaussian field) on the boundary of the tree, and the notion of weak convergence of probability measures for such random distributions, and what has to be proved from the point of view of the RG.
The latter is a discrete-time infinite-dimensional dynamical system corresponding to progressive coarse-graining or integration over high-momentum components of the field while keeping the low-momentum part fixed. The problem at hand, i.e., the UV limit, is related to the problem of conjugation of such a map to a Poincare-Dulac normal form. In order to gain insight into the RG dynamical system, we will do some heuristic calculations using formal perturbation theory, with close attention to the presence of resonant monomials, one of which is responsible for the logarithmic correction.
Finally, we will introduce a Brydges-Yau lift for the previous naive version of the RG map, which is more amenable to rigorous estimates while allowing for the crucial explicit extraction of the first two orders of perturbation theory. I will then introduce the norms required for the control of this lifted map, in particular for handling the so-called large-field problem. The main reference for the technical aspects of our main topic is the 2013 article jointly written by Ajay Chandra, Gianluca Guadagni, and the lecturer.