The Critical 2d Stochastic Heat Flow

Abstract

We consider directed polymers in random environment in the critical dimension $d=2$, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on ${\mathbb{R}}^2$ with logarithmic correlations, which we call the *Critical 2d Stochastic Heat Flow*. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

Minor revisions. To appear in Inventiones mathematicae

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