  After a sudden quench from the disordered high-temperature $T_0\to\infty$
phase to a final temperature below the critical point $T_F \ll T_c$, the
non-conserved order parameter dynamics of the two-dimensional ferromagnetic
Ising model on a square lattice (2dIM) initially approaches the critical
percolation state before entering the coarsening regime. This approach involves
two timescales associated with the first appearance (at time $t_{{\rm p}_1}>0$)
and stabilization (at time $t_{\rm p}>t_{{\rm p}_1}$) of a giant percolation
cluster, as previously reported. However, the microscopic mechanisms that
control such timescales are not yet fully understood. In this paper, in order
to study their role on each time regime after the quench ($T_F=0$), we
distinguish between spin flips that decrease the total energy of the system
from those that keep it constant, the latter being parametrized by the
probability $p$. We show that the cluster size heterogeneity $H(t,p)$ and the
typical domain size $\ell (t,p)$ have no dependence on $p$ in the first time
regime up to $t_{{\rm p}_1}$. On the other hand, the time for stabilizing a
percolating cluster is controlled by the acceptance probability of
constant-energy flips: $t_{\rm p}(p) \sim p^{-1}$ for $p\ll 1$ (at $p=0$, the
dynamics gets stuck in a metastable state). These flips are also the relevant
ones in the later coarsening regime where dynamical scaling takes place.
Because the phenomenology on the approach to the percolation point seems to be
shared by many 2d systems with a non-conserved order parameter dynamics (and
certain cases of conserved ones as well), our results may suggest a simple and
effective way to set, through the dynamics itself, $t_{{\rm p}_1}$ and $t_{\rm
p}$ in such systems.
