  The spontaneous symmetry breaking in the Ising model can be equivalently
described in terms of percolation of Wolff clusters. In O(n) spin models
similar clusters can be built in a general way, and they are currently used to
update these systems in Monte Carlo simulations. We show that for 3-dimensional
O(2), O(3) and O(4) such clusters are indeed the physical `islands' of the
systems, i.e., they percolate at the physical threshold and the percolation
exponents are in the universality class of the corresponding model. For O(2)
and O(3) the result is proven analytically, for O(4) we derived it by numerical
simulations.
