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We study the early time dynamics of bimodal spin systems on $2d$ latticesevolving with different microscopic stochastic updates. We treat theferromagnetic Ising model with locally conserved order parameter (Kawasakidynamics), the same model with globally conserved order parameter (nonlocalspin exchanges), and the voter model. As already observed for non-conservedorder parameter dynamics (Glauber dynamics), in all the cases in which thestochastic dynamics satisfy detailed balance, the critical percolation statepersists over a long period of time before usual coarsening of domains takesover and eventually takes the system to equilibrium. By studying thegeometrical and statistical properties of time-evolving spin clusters we areable to identify a characteristic length $\ell_p(t)$, different from the usuallength $\ell_d(t) \sim t^{1/z_{d}}$ that describes the late time coarsening,that is involved in all scaling relations in the approach to the criticalpercolation regime. We find that this characteristic length depends on theparticular microscopic dynamics and the lattice geometry. In the case of thevoter model, we observe that the system briefly passes through a criticalpercolation state, to later approach a dynamical regime in which the scalingbehaviour of the geometrical properties of the ordered domains can be ascribedto a different criticality.

Coarsening and percolation in the kinetic $2d$ Ising model with spin exchange updates and the voter model