  Stochastic motion of a point -- known as Brownian motion -- has many
successful applications in science, thanks to its scale invariance and
consequent universal features such as Gaussian fluctuations. In contrast, the
stochastic motion of a line, though it is also scale-invariant and arises in
nature as various types of interface growth, is far less understood. The two
major missing ingredients are: an experiment that allows a quantitative
comparison with theory and an analytic solution of the Kardar-Parisi-Zhang
(KPZ) equation, a prototypical equation for describing growing interfaces. Here
we solve both problems, showing unprecedented universality beyond the scaling
laws. We investigate growing interfaces of liquid-crystal turbulence and find
not only universal scaling, but universal distributions of interface positions.
They obey the largest-eigenvalue distributions of random matrices and depend on
whether the interface is curved or flat, albeit universal in each case. Our
exact solution of the KPZ equation provides theoretical explanations.
