Skeleton expansions for directed polymers in disordered media

Semjon Stepanow

Published 2007-09-30Version 1

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 $\Delta (t)$, 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 $\Delta (t)$ 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.