First-order transition in a percolation model with nucleation and preferential growth

The spanning cluster properties of a percolation model with nucleation and preferential growth exhibit first-order transitions depending on the values of the growth parameter $g_0$ and the initial seed concentration $\rho$. Except for the preferential growth of smaller clusters with a size-dependent growth probability of amplitude $g_0$, the model preserves all other criteria of the original percolation model. As $\rho$ decreases starting from the percolation threshold $p_c$ of the original percolation, a line of continuous transition encounters a coexistence region of percolative and nonpercolative large clusters. At sufficiently small values of $\rho$ ($\le 0.05$), the value of $g_0$ exceeds $p_c$ and generates compact spanning clusters leading to first-order discontinuous transitions.

Figure 1

The final static cluster configurations on a square lattice of size $L=128$ (a) at $g_{0c}\left(L\right)=0.169$ for $\rho =0.50$ and (b) at $g_{0c}\left(L\right)=0.751$ for $\rho =0.02$. The spanning clusters are shown in red and finite clusters are in different colors. The white space represents the inaccessible lattice sites.

Figure 2

Plot of ${\chi }_{\infty }/L^2$ vs $g_0$ for (a) $\rho =0.50$ and (b) $\rho =0.02$ for different lattice sizes $L=256(◯), 512(\square ), 1024(\diamond ), 2048\left(▵\right)$. Crosses on the $g_0$ axis represent the thresholds $g_{0c}\left(L\right)$. In the insets of (a) and (b), ${\chi }_{\max }$ is plotted against $L$. In (c), ${\chi }_{\infty }/L^{\beta /\nu \left(\rho \right)}$ is plotted against $[g_0-g_{0c}\left(L\right)]L^{1/\nu \left(\rho \right)}$ for $\rho =0.50$ (red) and $\rho =0.02$ (blue) using the same symbols for different $L$. In (d), $g_{0c}$ is plotted against $\rho$ for $L=2048$. In inset I of (d), the $L$ dependence of $g_{0c}$ is shown, and in inset II of (d), variations of $\gamma /\nu$ ($•$), $1/\nu$ ($▴$), and $d_f$ ($\blacksquare$) with $\rho$ are given.

Figure 3

Plot of $s_{\max }$ against $L$ (a) for $\rho =0.50$ (a) and (b) for $\rho =0.02$. $P_{\infty }{L}^{\beta /\nu \left(\rho \right)}$ vs $g_0$ and $P_{\infty }{L}^{\beta /\nu \left(\rho \right)}$ vs $[g_0-g_{0c}\left(L\right)]L^{1/\nu \left(\rho \right)}$ are plotted in insets I and II, respectively. The same symbol set of Fig. 2 is used for different $L$.

Figure 4

Plot of $B_{\rho ,L}\left(g_0\right)$ vs $g_0$ (a) for $\rho =0.50$ and (b) for $\rho =0.02$ for different $L$ using the same symbol set of Fig. 2. In the insets, $B_{\rho ,L}\left(g_0\right)$ is plotted against $[g_0-g_{0c}\left(L\right)]L^{1/\nu \left(\rho \right)}$. The dotted line in (a) represents the value of BC at threshold.

Figure 5

(a) Plot of $P_{\ell }\left(P_{\mathrm{large}}\right)$ against $P_{\mathrm{large}}$ for $L=2048$ for $\rho =0.01$ ($•$), 0.02 ($\blacksquare$), 0.03 ($⧫$), 0.05 ($▴$), 0.10 ($◂$), 0.42 ($▸$), 0.45 ($▾$), and 0.50 ($*$). In the inset of (a), $\mathrm{\Delta }{P}_{\mathrm{large}}$ is plotted against $L$ for several values of $\rho \le 0.05$. In (b), $P_c$ is plotted against $\rho$.