Abstract

We consider the directed polymer model on ${\mathbb{Z}}^d$, focusing on the critical dimension $d=2$. Our main contribution is to give a sharp lower bound on the free energy in the high-temperature regime. Our proof uses a percolation argument inspired by Lacoin (2010), for which we introduce a key property of bounded "log-energy": this property quantifies the regularity of the polymer measures at diffusive scales and we show that it propagates along open paths. Combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025), this gives upper and lower bounds on the free energy that are matching up to a multiplicative constant.

Organizer

Wai-Kit Lam (NTU)