Skeleton expansions for directed polymers in disordered media

Partial summations of perturbation expansions of the directed polymer in disordered media (DPRM) enables one to represent the latter as skeleton expansions in powers of the effective coupling constant $\mathrm{\Delta }(t)$, which corresponds to the binding state between two replicas in the replica field theory of DPRM, and is equivalent to the binding state of a quantum particle in an external $\delta$% -potential. The strong coupling phase is characterized by the exponential dependence of $\mathrm{\Delta }(t)$ on $t$, $\Delta (t)\sim \exp (p_{c}t)$with$% p_{c} $ being the binding energy of the particle. For dimensions $d>2$ the strong coupling phase exists for $\Delta_{0}>\Delta _{c}(d)$. We compute explicitly the mean-square displacement and the 2nd cumulant of the free energy to the lowest order in powers of effective coupling in $d=1$. We argue that the elimination of the terms $\exp (p_{c}t)$ in skeleton expansions demands an additional partial summation of skeleton series.