Directed polymers in random media under confining force

TL;DR

This paper investigates the scaling behavior of directed polymers in 2D random media under confining forces, proposing conjectures for their size and energy fluctuations, and confirms these through a novel exact ground state algorithm.

Contribution

It introduces a new algorithm with cubic time complexity to accurately find ground states and verifies theoretical scaling conjectures numerically.

Findings

Scaling exponents for radius of gyration and energy fluctuations are confirmed.

A novel algorithm efficiently computes exact ground states.

Theoretical conjectures are supported by numerical evidence.

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 $H\big(\{y(x)\}\big) = \sum_{x=1}^N \exyx + \epsilon \Wa^\alpha$, 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.