A geometric perspective on the scaling limits of critical Ising and $\varphi^4_d$ models

The lecture delivered at the \emph{Current Developments in Mathematics} conference (Harvard-MIT, 2021) focused on the recent proof of the Gaussian structure of the scaling limits of the critical Ising and $\varphi^4$ fields in the marginal case of four dimensions(joint work with Hugo Duminil-Copin). These notes expand on the background of the question addressed by this result, approaching it from two partly overlapping perspectives: one concerning critical phenomena in statistical mechanics and the other functional integrals over Euclidean spaces which could serve as a springboard to quantum field theory. We start by recalling some basic results concerning the models' critical behavior in different dimensions. The analysis is framed in the models' stochastic geometric random current representation. It yields intuitive explanations as well as tools for proving a range of dimension dependent results, including: the emergence in $2D$ of Fermionic degrees of freedom, the non-gaussianity of the scaling limits in two dimensions, and conversely the emergence of Gaussian behavior in four and higher dimensions. To cover the marginal case of $4D$ the tree diagram bound which has sufficed for higher dimensions needed to be supplemented by a singular correction. Its presence was established through multi-scale analysis in a recent joint work with HDC.