  In dimension $d \geq 3$, the directed polymer in a random medium undergoes a
phase transition between a free phase and a disorder dominated phase. For the
latter, Fisher and Huse have proposed a droplet theory based on the scaling of
the free energy fluctuations $\Delta F(l) \sim l^{\theta}$. On the other hand,
in related growth models belonging to the KPZ universality class, Forrest and
Tang have found that the height-height correlation function is logarithmic at
the transition. For the directed polymer model at criticality, this translates
into logarithmic free energy fluctuations $\Delta F_{T_c}(l) \sim (\ln
l)^{\sigma}$ with $\sigma=1/2$. In this paper, we propose a droplet scaling
analysis exactly at criticality based on this logarithmic scaling. Our main
conclusion is that the typical correlation length $\xi(T)$ of the low
temperature phase, diverges as $ \ln \xi(T) \sim (- \ln (T_c-T))^{1/\sigma}
\sim (- \ln (T_c-T))^{2} $. Furthermore, the logarithmic dependence of $\Delta
F_{T_c}(l)$ leads to the conclusion that the critical temperature $T_c$
actually coincides with the explicit upper bound $T_2$ derived by Derrida and
coworkers, where $T_2$ corresponds to the temperature below which the ratio
$\bar{Z_L^2}/(\bar{Z_L})^2$ diverges exponentially in $L$. Finally, since the
Fisher-Huse droplet theory was initially introduced for the spin-glass phase,
we briefly mention the similarities and differences with the directed polymer
model. If one speculates that the free energy of droplet excitations for
spin-glasses is also logarithmic at $T_c$, one obtains a logarithmic decay for
the mean square correlation function at criticality $\bar{C^2(r)} \sim 1/(\ln r
)^{\sigma}$.
