Zero temperature limit for (1+1) directed polymers with correlated random potential

TL;DR

This paper investigates the zero temperature limit of (1+1) directed polymers with correlated random potential, revealing a replica symmetry breaking transition and a crossover in free energy behavior at a characteristic temperature.

Contribution

It demonstrates the one-step replica symmetry breaking structure in the zero temperature limit for correlated potentials, and identifies a crossover temperature with distinct free energy regimes.

Findings

Identification of a crossover temperature T* ~ (u R)^{1/3}

High-temperature regime matches delta-correlated potential behavior

Low-temperature regime shows saturation of free energy prefactor

Abstract

Zero temperature limit in (1+1) directed polymers with finite range correlated random potential is studied. In terms of the standard replica technique it is demonstrated that in this limit the considered system reveals the one-step replica symmetry breaking structure similar to the one which takes place in the Random Energy Model. In particular, it is shown that at the temperature $T_{*} \sim (u R)^{1/3}$ (where $u$ and $R$ are the strength and the correlation length of the random potential) there is a crossover from the high- to the low-temperature regime. Namely, in the high-temperature regime at $T >> T_{*}$ the model is equivalent to the one with the $\delta$-correlated potential where the non-universal prefactor of the free energy is proportional to $T^{-2/3}$, while at $T << T_{*}$ this non-universal prefactor saturates at a finite (temperature independent) value.