Freezing transition of the directed polymer in a $1+d$ random medium : location of the critical temperature and unusual critical properties

In dimension $d\ge 3$, 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 $\mathrm{\Delta }F(l)\sim l^{\theta }$. 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} $\mathrm{.}Furthermore,thelogarithmicdependenceof$\Delta F_{T_c}(l)$leadstotheconclusionthatthecriticaltemperature$T_c$actuallycoincideswiththeexplicitupperbound$T_2$ derived by Derrida and coworkers, where $T_2$ corresponds to the temperature below which the ratio $\overline{Z_L^2}\mathrm{/}(\overline{Z_L}{)}^2$ 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 $\bar{C^2®} \sim 1/(\ln r )^{\sigma}$.