  Despite a long history and a clear overall understanding of properties of
random walks on an incipient infinite cluster in percolation, some important
information on it seems to be missing in the literature. In the present work,
we revisit the problem by performing massive numerical simulations for
(sub)diffusion of particles on such clusters. Thus, we discuss the shape of the
probability density function (PDF) of particles' displacements, and the way it
converges to its long-time limiting scaling form. Moreover, we discuss the
properties of the mean squared displacement (MSD) of a particle diffusing on
the infinite cluster at criticality. This one is known not to be
self-averaging. We show that the fluctuations of the MSD in different
realizations of the cluster are universal, and discuss the properties of the
distribution of these fluctuations. These strong fluctuations coexist with the
ergodicity of subdiffusive behavior in the time domain. The dependence of the
relative strength of fluctuations in time-averaged MSD on the total trajectory
length (total simulation time) is characteristic for diffusion in a percolation
system and can be used as an additional test to distinguish this process with
disorder-induced memory from processes with otherwise similar behavior, like
fractional Brownian motion with the same value of the Hurst exponent.
