When $d\ge 3$, the directed polymer a in random environment on $\mathbb Z^d$ is known to display a phase transition from a diffusive phase, known as \textit{weak disorder} to a localized phase, referred to as \textit{strong disorder}. This transition is encoded by the behavior of the the free energy of the model, defined by
$$\mathfrak f(\beta):=\lim_{N\to \infty} (1/n)\log W^{\beta}_n$$
where $W^{\beta}_n$ is the normalized partition function for the directed polymer of length $n$. More precisely weak disorder corresponds to $\mathfrak f(\beta)=0$ and strong disorder to $\mathfrak f(\beta)<0$. Monotonicity and continuity of $\mathfrak f$ implies that there exists $\beta_c\in [0,\infty]$ such that weak disorder is equivalent to $\beta\in [0,\beta_c]$. Furthermore $\beta_c>0$ if and only if $d\ge 3$.
We will present recent bounds on the free-energy obtained near critically. More precisely
$$ \exp( u^{-1+o(1)}) \mathfrak f(\beta_c+u)\le \exp(- u^{-\gamma(d)+o(1)})$$
where $\gamma(d)=\frac{d+2}{8d}$.