The Critical 2d Stochastic Heat Flow and Related Models

TL;DR

This paper reviews recent advances in understanding the critical 2d stochastic heat flow, highlighting phase transitions, Gaussian fluctuations, and the novel scaling limit that extends solution theories for singular SPDEs in two dimensions.

Contribution

It introduces the critical 2d stochastic heat flow as a new scaling limit, providing a framework to define solutions in the critical dimension beyond existing theories.

Findings

Identification of a phase transition at intermediate disorder scale

Gaussian fluctuations in the sub-critical regime

Existence of a unique critical scaling limit for the 2d stochastic heat equation

Abstract

In these lecture notes, we review recent progress in the study of the stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. It was discovered that a phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and named the critical 2d stochastic heat flow. This gives a meaning to the solution of the stochastic heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems and singular SPDEs. A list of open questions is also provided.