Quentin Berger

2D directed polymers and the Stochastic Heat Flow

I will review some recent results on the directed polymer model in dimension d=2, which can be seen as a discretised version of the stochastic heat equation. Recent results have shown that, in a suitable (and subtle) critical window of parameters, the model converges to a disordered limit, namely the Critical 2D Stochastic Heat Flow (SHF) introduced by Caravenna, Sun and Zygouras. This SHF is a non-Gaussian stochastic process of measures on R^2, and a lot of its properties have been studied over the last years. In this talk, I will first take some time to explain the (idea of the) construction of the SHF and then review some of the recent advances. In particular, I will comment on recentworks in collaboration with F. Caravenna, N. Turchi and N. Zygouras, which study how the mass assigned by the SHF to a ball behaves, either as time goes to infinity or as the radius of the ball goes to 0.