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 $\omega=(\omega_{n,x})_{n\geq 0,x\in\mathbb Z^d}$, 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 $$ -\mathfrak{f}(\beta) \asymp \exp{\Big(- \frac{\pi}{\sigma^2(\beta)}\Big)},\quad \text{ as $\beta\downarrow 0$.} $$