Skeleton expansions for directed polymers in disordered media

TL;DR

This paper develops skeleton expansions for directed polymers in disordered media, analyzing the strong coupling phase and computing key statistical measures in one dimension, with implications for understanding binding states and phase transitions.

Contribution

It introduces a method to represent DPRM perturbation series as skeleton expansions and analyzes the strong coupling phase, including explicit calculations in one dimension.

Findings

Strong coupling phase characterized by exponential growth of effective coupling

Explicit calculation of mean-square displacement in 1D

Identification of the need for additional summation to eliminate exponential terms

Abstract

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.