Directed polymers in a random medium: a variational approach

A disorder-dependent Gaussian variational approach is applied to the problem of a $d$ dimensional polymer chain in a random medium (or potential). Two classes of variational solutions are obtained. For $d<2$, these two classes may be interpreted as domain and domain wall. The critical exponent $\nu$ describing the polymer width is $\nu={1\over (4-d)}$ (domain solution) or $\nu={3\over (d+4)}$ (domain wall solution). The domain wall solution is equivalent to the (full) replica symmetry breaking variational result. For $d>2$, we find $\nu={1\over 2}$. No evidence of a phase transition is found for $2< d< 4$: one of the variational solutions suggests that the polymer chain breaks into Imry-Ma segments, whose probability distribution is calculated. For $d>4$, the other variational solution undergoes a phase transition, which has some similarity with B. Derrida's random energy models.