After a sudden quench from the disordered high-temperature $T_0\to \infty$ phase to a final temperature well below the critical point $T_{\mathrm{F}}\ll T_c$, the nonconserved order parameter dynamics of the two-dimensional ferromagnetic Ising model on a square lattice initially approaches the critical percolation state before entering the coarsening regime. This approach involves two timescales associated with the first appearance (at time $t_{{\mathrm{p}}_1}>0$) and stabilization (at time $t_{\mathrm{p}}>t_{{\mathrm{p}}_1}$) of a giant percolation cluster, as previously reported. However, the microscopic mechanisms that control such timescales are not yet fully understood. In this paper, to study their role on each time regime after the quench ($T_{\mathrm{F}}=0$), 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 observables such as 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 $t_{{\mathrm{p}}_1}$. Furthermore, when energy-decreasing flips are forbidden while allowing constant-energy flips, the kinetics is essentially frozen after the quench and there is no percolation event whatsoever. Taken together, these results indicate that the emergence of the first percolating cluster at $t_{{\mathrm{p}}_1}$ is completely driven by energy decreasing flips. However, the time for stabilizing a percolating cluster is controlled by the acceptance probability of constant-energy flips: $t_{\mathrm{p}}\left(p\right)\sim p^{-1}$ 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 nonconserved 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, $t_{{\mathrm{p}}_1}$ and $t_{\mathrm{p}}$ in such systems.

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• Nonequilibrium statistical mechanics
• Percolation