We present results of our numerical study of the critical dynamics of percolation observables for the two-dimensional Ising model. We consider the (Monte Carlo) short-time evolution of the system with small initial magnetization and heat-bath dynamics. We find qualitatively different dynamic behaviors for the magnetization M and for omega, the so-called strength of the percolating cluster, which is the order parameter of the percolation transition. More precisely, we obtain a (leading) exponential form for omega as a function of the Monte Carlo time t, to be compared with the power-law increase encountered for M at short times. Our results suggest that, although the descriptions in terms of magnetic or percolation order parameters may be equivalent in the equilibrium regime, greater care must be taken to interpret percolation observables at short times.
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point
Tc of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability
pB<1, the corresponding site-bond clusters keep on percolating at
Tc and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability
pc such that, for any
pB> pc, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for
pc<pB< 1 but, for
pB=pc, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.
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 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.