Directed polymers in random media under confining force

Abstract

The scaling behavior of a directed polymer in a two-dimensional (2D) random potential under confining force is investigated. The energy of a polymer with configuration $\{y(x)\}$ is given by $H\big(\{y(x)\}\big) = \sum_{x=1}^N \exyx + ε\Wa^α$, where $\eta (x,y)$ is an uncorrelated random potential and $\Wa$ is the width of the polymer. Using an energy argument, it is conjectured that the radius of gyration $R_g(N)$ and the energy fluctuation $\Delta E(N)$ of the polymer of length $N$ in the ground state increase as $R_g(N)\sim N^{\nu }$ and $\Delta E(N)\sim N^{\omega }$ respectively with $\nu =1/(1+\alpha )$ and $\omega =(1+2\alpha )/(4+4\alpha )$ for $\alpha \ge 1/2$. A novel algorithm of finding the exact ground state, with the effective time complexity of $\cO(N^3)$, is introduced and used to confirm the conjecture numerically.

9 pages, 7 figures

References in corpus (4)

• Shape Fluctuations and Random Matrices
• Phase Diagram of Optimal Paths
• Anomalous Roughness, Localization, and Globally Constrained Random Walks
• Fluctuations of self-flattening surfaces