The circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer

Abstract

We consider the Kardar-Parisi-Zhang (KPZ) equation for a circular interface in two dimensions, unconstrained by the standard small-slopes and no-overhang approximations. Numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short-time regime with the interface fluctuations of a self-avoiding ring or 2D vesicle, and a long-time regime governed by the Tracy-Widom distribution expected for this geometry. For small noise amplitudes, scaling behavior is only of the latter type. Large noise is also seen to renormalize the bare physical parameters of the ring, akin to analogous parameter renormalization for equilibrium 3D membranes. Our results bear particular importance on the relation between relevant universality classes of scale-invariant systems in two dimensions.