From disordered systems to the Critical 2D Stochastic Heat Flow

TL;DR

This paper reviews the development of the Critical 2D Stochastic Heat Flow, a universal non-Gaussian process solving a critical singular SPDE in two dimensions, linking disorder relevance to SPDE criticality.

Contribution

It introduces the Critical 2D Stochastic Heat Flow, a novel universal process that addresses the critical dimension for the stochastic heat equation, advancing understanding of disordered systems and singular SPDEs.

Findings

Construction of the Critical 2D Stochastic Heat Flow.

Provides a non-Gaussian scaling limit at criticality.

Links disorder relevance to SPDE sub/super-criticality.

Abstract

We review our joint work on the scaling limits of disordered systems, linking the notion of disorder relevance/irrelevance to that of sub/super-criticality of singular SPDEs. This line of research culminated in the construction of the Critical 2D Stochastic Heat Flow (SHF), a universal process which provides a non-trivial solution to the Stochastic Heat Equation in dimension 2, a critical singular SPDE that lies beyond the reach of existing solution theories. The SHF also offers a rare example of a non-Gaussian scaling limit for a disordered system at its phase transition point in the critical dimension.