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.
