Stochastic evolution of interacting directed strings

We study the stochastic dynamics of a system of $N$ polymer rings on an infinite cylinder with only excluded volume interaction, using simulations and also analytically. The strings can diffuse in the transverse direction by local Rouse-like moves where an up-kink changes to a down -kink, or vice versa. The motion is biased if the rates of flip are not equal. In the initial state, we start with a step initial condition: all the sites with a positive $y$ coordinate are empty, and the sites with $y \leq 0$ are occupied with a uniform density. We study the scaling form of the density profile in the hydrodynamic limit. The scaling form is a function of $y/\sqrt{t}$ in the unbiased case, and a function of $y/t$ in the biased case. In the unbiased case, in the steady state on a torus, the marginal probability distribution of the positions of the strings on a line with fixed $x$-coordinate is exactly same as that in the ground state of quantum-mechanical spin-less fermions on a line. The biased problem is equivalent to a growing surface in $2+1$ dimension, which is expected to be in the Kardar-Parisi-Zhang universality class.