  We study electron transport at the edge of a generic disordered
two-dimensional topological insulator, where some channels are topologically
protected from backscattering. Assuming the total number of channels is large,
we consider the edge as a quasi-one-dimensional quantum wire and describe it in
terms of a non-linear sigma model with a topological term. Neglecting
localization effects, we calculate the average distribution function of
transmission probabilities as a function of the sample length. We mainly focus
on the two experimentally relevant cases: a junction between two quantum Hall
(QH) states with different filling factors (unitary class) and a relatively
thick quantum well exhibiting quantum spin Hall (QSH) effect (symplectic
class). In a QH sample, the presence of topologically protected modes leads to
a strong suppression of diffusion in the other channels already at the scales
much shorter than the localization length. On the semiclassical level, this is
accompanied by the formation of a gap in the spectrum of transmission
probabilities close to unit transmission, thereby suppressing shot noise and
conductance fluctuations. In the case of a QSH system, there is at most one
topologically protected edge channel leading to weaker transport effects. In
order to describe `topological' suppression of nearly perfect transparencies,
we develop an exact mapping of the semiclassical limit of the one-dimensional
sigma model onto a zero-dimensional sigma model of a different symmetry class,
allowing us to identify the distribution of transmission probabilities with the
average spectral density of a certain random-matrix ensemble. We extend our
results to other symmetry classes with topologically protected edges in two
dimensions.
