Cluster percolation in O(n) spin models

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. Wt show that for three-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.