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We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ)universality class with curved initial conditions. By simulations of a clustergrowth model and experiments of liquid-crystal turbulence, we determine theuniversal scaling functions that describe the height distribution and thespatial correlation of the interfaces growing outward from a ring. The scalingfunctions, controlled by a single dimensionless time parameter, show crossoverfrom the statistical properties of the flat interfaces to those of the circularinterfaces. Moreover, employing the KPZ variational formula to describe thecase of the ring initial condition, we find that the formula, which wenumerically evaluate, reproduces the numerical and experimental resultsprecisely without adjustable parameters. This demonstrates that precisenumerical evaluation of the variational formula is possible at all, andunderlines the practical importance of the formula, which is able to predictthe one-point distribution of KPZ interfaces for general initial conditions.

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