Sharp behavior of the free energy for the two-dimensional directed polymer model

We consider the directed polymer model on ${\mathbb{Z}}^d$, in an i.i.d.\ random environment $\omega =({\omega }_{n,x}{)}_{n\ge 0,x\in {\mathbb{Z}}^d}$, 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. Writing $\mathfrak{f}(\beta )$ for the quenched free energy, and setting $\lambda (\beta ):=\log \mathbb{E}[e^{\beta {\omega }_{1,0}}]$ and $\sigma (\beta {)}^2:=e^{\lambda (2\beta )-2\lambda (\beta )}-1$, our lower bound combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025) gives $$-\mathfrak{f}(\beta )\asymp \exp (-\frac{\pi }{{\sigma }^2(\beta )}),\text{ as }\beta \downarrow 0\text{.}$$