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 $η(x,y)$ 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 $ΔE(N)$ of the polymer of length $N$ in the ground state increase as $R_g(N)\sim N^ν$ and $ΔE(N)\sim N^ω$ respectively with $ν= 1/(1+α)$ and $ω= (1+2α)/(4+4α)$ for $α\ge 1/2$. 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.