  We provide a comprehensive report on scale-invariant fluctuations of growing
interfaces in liquid-crystal turbulence, for which we recently found evidence
that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1
dimensions [Phys. Rev. Lett. 104, 230601 (2010); Sci. Rep. 1, 34 (2011)]. Here
we investigate both circular and flat interfaces and report their statistics in
detail. First we demonstrate that their fluctuations show not only the KPZ
scaling exponents but beyond: they asymptotically share even the precise forms
of the distribution function and the spatial correlation function in common
with solvable models of the KPZ class, demonstrating also an intimate relation
to random matrix theory. We then determine other statistical properties for
which no exact theoretical predictions were made, in particular the temporal
correlation function and the persistence probabilities. Experimental results on
finite-time effects and extreme-value statistics are also presented. Throughout
the paper, emphasis is put on how the universal statistical properties depend
on the global geometry of the interfaces, i.e., whether the interfaces are
circular or flat. We thereby corroborate the powerful yet geometry-dependent
universality of the KPZ class, which governs growing interfaces driven out of
equilibrium.
