The Wandering Exponent of a One-Dimensional Directed Polymer in a Random Potential with Finite Correlation Radius

TL;DR

This paper investigates the wandering behavior of a one-dimensional directed polymer in a Gaussian random potential with finite correlation length, revealing a temperature-dependent transition in the solution and confirming a universal wandering exponent at low temperatures.

Contribution

It extends the understanding of directed polymers by deriving a new low-temperature solution with one-step replica symmetry breaking for finite correlation length potentials.

Findings

Kardar's solution applies only at high temperatures.

New low-temperature solution with replica symmetry breaking is obtained.

Wandering exponent z = 2/3 is confirmed at low temperatures.

Abstract

We consider a one-dimensional directed polymer in a random potential which is characterized by the Gaussian statistics with the finite size local correlations. It is shown that the well-known Kardar's solution obtained originally for a directed polymer with delta-correlated random potential can be applied for the description of the present system only in the high-temperature limit. For the low temperature limit we have obtained the new solution which is described by the one-step replica symmetry breaking. For the mean square deviation of the directed polymer of the linear size L it provides the usual scaling $L^{2z}$ with the wandering exponent z = 2/3 and the temperature-independent prefactor.