Freezing transition of the directed polymer in a 1 + d random medium: location of the critical temperature and unusual critical properties.
In dimension d > or =3, the directed polymer in a random medium undergoes a phase transition between a free phase at high temperature and a low-temperature disorder-dominated phase. For the latter phase, Fisher and Huse have proposed a droplet theory based on the scaling of the free-energy fluctuations Delta F(l) approximately l theta at scale l. On the other hand, in related growth models belonging to the Kardar-Parisi-Zhang 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 FTc(l) approximately (lnl)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) approximately [-ln(Tc-T)]1/sigma approximately [-ln(Tc-T)]2, instead of the usual power law xi(T) approximately (Tc-T)-v. Furthermore, the logarithmic dependence of Delta FTc(l) leads to the conclusion that the critical temperature Tc actually coincides with the explicit upper bound T2 derived by Derrida and co-workers, where T2 corresponds to the temperature below which the ratio Z 2 L/(ZL)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 with and differences from the directed polymer model. If one speculates that the free energy of droplet excitations for spin glasses is also logarithmic at Tc, one obtains a logarithmic decay for the mean square correlation function at criticality, C2(r) approximately 1/(ln r)sigma, instead of the usual power law l/rd-2+eta.
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Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS, 91191 Gif-sur-Yvette cedex, France.
Garel
Thomas
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