  We study the topological charge distribution of the SU(3) Yang--Mills theory
with high precision in order to be able to detect deviations from Gaussianity.
The computation is carried out on the lattice with high statistics Monte Carlo
simulations by implementing a naive discretization of the topological charge
evolved with the Yang--Mills gradient flow. This definition is far less
demanding than the one suggested from Neuberger's fermions and, as shown in
this paper, in the continuum limit its cumulants coincide with those of the
universal definition appearing in the chiral Ward identities. Thanks to the
range of lattice volumes and spacings considered, we can extrapolate the
results for the second and fourth cumulant of the topological charge
distribution to the continuum limit with confidence by keeping finite volume
effects negligible with respect to the statistical errors. Our best results for
the topological susceptibility is t_0^2*chi=6.67(7)*10^-4, where t_0 is a
standard reference scale, while for the ratio of the forth cumulant over the
second we obtain R=0.233(45). The latter is compatible with the expectations
from the large Nc expansion, while it rules out the theta-behavior of the
vacuum energy predicted by the dilute instanton model. Its large distance from
1 implies that, in the ensemble of gauge configurations that dominate the path
integral, the fluctuations of the topological charge are of quantum
non-perturbative nature.
