Tricritical Point in Explosive Percolation

The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent $1/{\phi }_t=1.8\pm 0.1$.

Figure 1

Size dependence, for the best-of-ten rule ($q=0$), of the maximum in the susceptibility $\chi$ and the order parameter $P_{\infty }$ (inset), at the percolation transition, for square lattices with size $L^2$, with $L$ ranging from 32 to 4096 sites. For large system sizes, the order parameter converges to a constant value and the susceptibility scales with $L^d$, where $d$ is the dimension of the system. Results have been averaged over ${10}^4$ samples. Error bars are smaller than 0.6%.

Figure 2

Order parameter ($P_{\infty }$) as a function of the fraction of occupied bonds ($p$), for different values of the degree of dilution $q$. Two different regimes are found. For lower values of $q$ the transition is discontinuous while for larger values it is continuous. Increasing $q$ the percolation threshold goes from the one for the best-of-ten product rule to the one for classical percolation. Results have been averaged over ${10}^4$ samples for square lattices with lateral size of 4096, and the error bars are below 0.2%.

Figure 3

Phase diagram of the model. The horizontal axis is the degree of dilution $q$ and the vertical one the fraction of occupied bonds $p$. The (green) dashed line is the first-order transition line and the (red) solid line is the continuous transition. Both lines meet at a tricritical point ($q_t$, $p_t$). Below the transition lines, there is no infinite cluster in the system. Above the transition lines an infinite cluster exists. The (blue) arrows stand for the relevant scaling fields ${\mu }_q$ and ${\mu }_p$ (discussed in the text). The values have been obtained from the crossing of the wrap probability, with error bars below 0.2%.

Figure 4

Scaling of the critical line, according to Eq. (1), for different values of $q$. A straight line is obtained for the tricritical value $q_t=0.51\pm 0.01$. The obtained crossover exponent ${\phi }_t$ is such that $1/{\phi }_t=1.8\pm 0.1$.

Figure 5

Size dependence of the maximum in the susceptibility ($\chi$) and for the order parameter ($P_{\infty }$), at the tricritical point of the hybrid model. System sizes range from 32 to 4096 lateral sites. Results have been averaged over ${10}^5$ samples. Error bars are smaller than the point size (below 0.5%).

Figure 6

(a) Finite-size scaling for the susceptibility ($\chi$) for $q=q_t$, according to the scaling ansatz of Eq. (3). Results for square lattices with lateral size of 1024, 2048, and 4096 sites, are averaged over ${10}^4$ samples. The value of ${\gamma }_t=2.9\pm 0.1$ can be obtained from the slope of the dotted line. (b) Crossover scaling for the susceptibility ($\chi$) for different values of $q=\{0.515,0.52,0.53,0.54,0.55\}$ and the largest system size (${4096}^2$ sites). Dashed and dotted lines correspond to the two different regimes (details in the text). Results have been averaged over ${10}^4$ samples of a square lattice with 4096 sites of lateral size.