The localization transition for the directed polymer in a random environment is smooth

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 $\mathit{\text{weak}}\mathit{\text{disorder}}$ to a localized phase, referred to as $\mathit{\text{strong}}\mathit{\text{disorder}}$. This transition is encoded by the behavior of the the free energy of the model, defined by $$\mathfrak{f}(\beta ):=\underset{N\to \mathrm{\infty }}{\lim }(1\mathrm{/}n)\log W_n^{\beta }$$ where $W_n^{\beta }$ 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 ${\beta }_c\in [0,\mathrm{\infty }]$ such that weak disorder is equivalent to $\beta \in [0,{\beta }_c]$. Furthermore ${\beta }_c>0$ if and only if $d\ge 3$. We prove that this transition is infinitely smooth in the sense that $\mathfrak{f}$ grows slower than any power function at the vicinity of ${\beta }_c$, that is $$\underset{\beta \downarrow {\beta }_c}{\lim }\frac{\log \mathrm{\mid }\mathfrak{f}(\beta )\mathrm{\mid }}{\log (\beta -{\beta }_c)}=\mathrm{\infty }\mathrm{.}$$