Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula

TL;DR

This paper investigates the fluctuations of KPZ class interfaces with curved initial shapes through simulations and experiments, demonstrating the accuracy of a variational formula in predicting height distributions and correlations.

Contribution

It introduces a variational formula for KPZ interfaces with curved initial conditions and validates it against simulations and experiments, showing precise predictions without adjustable parameters.

Findings

Universal scaling functions describe height distribution and spatial correlation.

Crossover observed from flat to circular interface statistics.

Variational formula accurately reproduces experimental and numerical results.

Abstract

We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved initial conditions. By simulations of a cluster growth model and experiments of liquid-crystal turbulence, we determine the universal scaling functions that describe the height distribution and the spatial correlation of the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionless time parameter, show crossover from the statistical properties of the flat interfaces to those of the circular interfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initial condition, we find that the formula, which we numerically evaluate, reproduces the numerical and experimental results precisely without adjustable parameters. This demonstrates that precise numerical evaluation of the variational formula is possible at all, and underlines the practical importance of the formula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.