Cluster percolation and critical behaviour in spin models and SU(N) gauge theories

The critical behaviour of several spin models can be simply described as percolation of some suitably defined clusters or droplets: the onset of the geometrical transition coincides with the critical point, and the percolation exponents are equal to the thermal exponents. It is still unknown whether, given a model, one can define at all the droplets. In the cases where this is possible, the droplet definition depends in general on the specific model at study and can be quite involved. We propose here a simple general definition for the droplets: they are clusters obtained by joining nearest-neighbour spins of the same sign with some bond probability p(CK), which is the minimal probability that still allows the existence of a percolating cluster at the critical temperature T-c. This definition has been recently introduced and indeed satisfies the. conditions required for the droplets, for many classical spin models, discrete and continuous, in two dimensions. By means of lattice Monte Carlo simulations we show here that the definition works as well in three dimensions. In particular, our prescription allows us to describe exactly the confinement-deconfinement transition of SU(N) gauge theories as Polyakov loop percolation.