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• W1966587746 abstract "In dimension $densuremath{geqslant}3$, 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 $ensuremath{Delta}F(l)ensuremath{sim}{l}^{ensuremath{theta}}$ 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 $ensuremath{Delta}{F}_{{T}_{c}}(l)ensuremath{sim}{(mathrm{ln}phantom{rule{0.2em}{0ex}}l)}^{ensuremath{sigma}}$ with $ensuremath{sigma}=1∕2$. 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 $ensuremath{xi}(T)$ of the low-temperature phase diverges as $mathrm{ln}phantom{rule{0.2em}{0ex}}ensuremath{xi}(T)ensuremath{sim}{[ensuremath{-}mathrm{ln}({T}_{c}ensuremath{-}T)]}^{1∕ensuremath{sigma}}ensuremath{sim}{[ensuremath{-}mathrm{ln}({T}_{c}ensuremath{-}T)]}^{2}$, instead of the usual power law $ensuremath{xi}(T)ensuremath{sim}{({T}_{c}ensuremath{-}T)}^{ensuremath{-}ensuremath{nu}}$. Furthermore, the logarithmic dependence of $ensuremath{Delta}{F}_{{T}_{c}}(l)$ leads to the conclusion that the critical temperature ${T}_{c}$ actually coincides with the explicit upper bound ${T}_{2}$ derived by Derrida and co-workers, 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 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 ${T}_{c}$, one obtains a logarithmic decay for the mean square correlation function at criticality, $overline{{C}^{2}(r)}ensuremath{sim}1∕{(mathrm{ln}phantom{rule{0.2em}{0ex}}r)}^{ensuremath{sigma}}$, instead of the usual power law $1∕{r}^{densuremath{-}2+ensuremath{eta}}$." @default.
• W1966587746 created "2016-06-24" @default.
• W1966587746 creator A5060853119 @default.
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• W1966587746 date "2006-07-05" @default.
• W1966587746 modified "2023-09-25" @default.
• W1966587746 title "Freezing transition of the directed polymer in a1+drandom medium: Location of the critical temperature and unusual critical properties" @default.
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• W1966587746 doi "https://doi.org/10.1103/physreve.74.011101" @default.
• W1966587746 hasPubMedId "https://pubmed.ncbi.nlm.nih.gov/16907055" @default.
• W1966587746 hasPublicationYear "2006" @default.
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