Cluster percolation in O(n) spin models

The spontaneous symmetry breaking in the Ising model can be equivalently de scribed in terms of percolation of Wolff clusters. In O(n) spin models simi lar clusters can be built in a general way, and they are currently used to update these systems in Monte Carlo simulations. Wt show that for three-dim ensional 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 per colation 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.