Localization and freezing of a Gaussian chain in a quenched random potential

The Gaussian chain in a quenched random potential (which is characterized by the disorder strength $\mathrm{\Delta }$) is investigated in the $d$ - dimensional space by the replicated variational method. The general expression for the free energy within so called one - step - replica symmetry breaking (1 - RSB) scenario has been systematically derived. We have shown that the replica symmetrical (RS) limit of this expression can describe the chain center of mass localization and collapse. The critical disorder when the chain becomes localized scales as ${\mathrm{\Delta }}_c\simeq b^d{N}^{-2+d\mathrm{/}2}$ (where $b$ is the length of the Kuhn segment length and $N$ is the chain length) whereas the chain gyration radius $R_{\mathrm{g}}\simeq b(b^d\mathrm{/}\mathrm{\Delta }{)}^{1\mathrm{/}(4-d)}$. The freezing of the internal degrees of freedom follows to the 1-RSB - scenario and is characterized by the beads localization length $\overline{{\mathcal{D}}^2}$. It was demonstrated that the solution for $\overline{{\mathcal{D}}^2}$ appears as a metastable state at $\mathrm{\Delta }={\mathrm{\Delta }}_A$ and behaves similarly to the corresponding frozen states in heteropolymers or in $p$ - spin random spherical model.