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 ν describing the polymer width is ν = 1 (4−d) (domain solution) or ν = 3 (d+4) (domain wall solution). The domain wall solution is equivalent to the (full) replica symmetry breaking variational result. For d > 2, we find ν = 12 . 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.