The critical 2d stochastic heat flow

We study the 2d Stochastic Heat Equation, that is the heat equation in two space dimensions with a multiplicative random potential (space-time white noise). This equation is ill-defined due to the singularity of the potential and we regularise it by discretising space-time, so that the solution can be identified with the partition function of a statistical mechanics model, the so-called directed polymer in random environment. We prove that, as discretisation is removed and the noise strength is rescaled in a critical way, the solution has a well-defined and unique limit: a universal process of random measures on R^2, which we call the critical 2d Stochastic Heat Flow. We investigate its features, showing that it cannot be the exponential of a generalised Gaussian field.

(joint work with R. Sun and N. Zygouras)