  The properties of the pure-site clusters of spin models, i.e. the clusters
which are obtained by joining nearest-neighbour spins of the same sign, are
here investigated. In the Ising model in two dimensions it is known that such
clusters undergo a percolation transition exactly at the critical point. We
show that this result is valid for a wide class of bidimensional systems
undergoing a continuous magnetization transition. We provide numerical evidence
for discrete as well as for continuous spin models, including SU(N) lattice
gauge theories. The critical percolation exponents do not coincide with the
ones of the thermal transition, but they are the same for models belonging to
the same universality class.
