  Dynamics of Wilson loops in pure Yang-Mills theories is analyzed in terms of
random walks of the holonomies of the gauge field on the gauge group manifold.
It is shown that such random walks should necessarily be free. The distribution
of steps of these random walks is related to the spectrum of string tensions of
the theory and to certain cumulants of Yang-Mills curvature tensor. It turns
out that when colour charges are completely screened, the holonomies of the
gauge field can change only by the elements of the group center, which
indicates that in the screening regime confinement persists due to thin center
vortices. Thick center vortices are also considered and the emergence of such
stepwise changes in the limits of infinitely thin vortices and infinitely large
loops is demonstrated.
