Thanks to the theories of Regularity Structures, Paracontrolled Distributions
and Renormalisation we now have a robust framework for singular SPD Es, which are “sub-critical”
in the sense of renormalisation. Recently, there have been efforts to approach the situation
of “critical” SPD Es and statistical mechanics models. A first such treatment has been
through the study of the two-dimensional stochastic heat equation, which has revealed a
certain phase transition and has led to the construction of the novel object called the
Critical 2d Stochastic Heat Flow. In this talk we will present some aspects of this model
and its construction. We will also present developments relating to other critical SPD Es.
Parts of this talk are based on joint works with Caravenna and Sun and others with Rosati and Gabriel.