  We consider the averaging process on the discrete $d$-dimensional torus. On
this graph, the process is known to converge to equilibrium on diffusive
timescales, not exhibiting cutoff. In this work, we refine this picture in two
ways. Firstly, we prove a concentration phenomenon of the averaging process
around its mean, occurring on a shorter timescale than the one of its
relaxation to equilibrium. Secondly, we establish sharp gradient estimates,
which capture its fast local smoothness property. This is the first setting in
which these two features of the averaging process -- concentration and local
smoothness -- can be quantified. These features carry useful information on a
number of large scale properties of the averaging process. As an illustration
of this fact, we determine the limit profile of its distance to equilibrium and
derive a quantitative hydrodynamic limit for it. Finally, we discuss their
implications on cutoff for the binomial splitting process, the particle
analogue of the averaging process.
