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

Abstract

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 $\Delta F(l)\sim l^{\theta }$. 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 $\overline{Z_L^2}/(\overline{Z_L}{)}^2$ 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 $\overline{C^2(r)}\sim 1/(\ln r{)}^{\sigma }$.

arXiv commentfinal version, 16 pages, Appendix added

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