  In the spirit of recent works on topological chaos generated by sequential
rotation of infinitely thin stirrers placed in a viscous liquid, we consider
the statistical properties of braiding exponent which quantitatively
characterizes the chaotic behavior of advected particles in two-dimensional
flows. We pay a special attention to the random stirring protocol and study the
time-dependent behavior of the variance of the braiding exponent. We show that
this behavior belongs to the Kardar-Parisi-Zhang universality class typical for
models of nonstationary growth. Using the matrix (Magnus) representation of the
braid group generators, we relate the random stirring protocol with the growth
of random heap generated by a ballistic deposition.
