Largest cluster in percolation: Implications for fragmentation studies

The probability distribution of the largest cluster size is studied within a three-dimensional bond percolation model on small lattices. Cumulants of the distribution exhibit distinct features near the percolation transition (pseudocritical point), providing a method for its identification. The location of the critical point in the continuous limit can be estimated without variation of the system size. This method is remarkably insensitive to finite-size effects and may be applied even for a very small system. The possibility of using various measurable quantities for sorting events makes the procedure useful in studying clusterization phenomena, in particular nuclear multifragmentation. Finite-size scaling and Δ-scaling relations are examined. The role of surface effects is evaluated by a comparison of results for free and periodic boundary conditions.

Figure 1

Bond percolation on a lattice of linear size $L=5$ with free boundary conditions. Plotted as a function of the bond probability: (a) the probability that a lattice is spanned, (b) moments of the average distribution of cluster sizes, (c) the variance of the largest cluster size distribution with various normalizations. (d) The average distribution of cluster sizes.

Figure 2

Probability distributions of the largest cluster size for a lattice of size $N=125$ with free boundary conditions.

Figure 3

The cumulant ratios of Eqs. (2) as functions of (a) the bond probability and (b) the scaling variable for systems of different sizes. Calculations are made with free boundary conditions.

Figure 4

Finite-size scaling plot for values of the bond probability p at which the conditions indicated on the figure are fulfilled. The line is the best linear fit to the open circles.

Figure 5

Same as Fig. 3 but for periodic boundary conditions. The filled circles in (a) indicate the $K_3$ and $K_4$ values for the Gumbel distribution (see text).

Figure 6

The normalized variance of $P(s_{\max })$ versus the bond probability for (a) free boundary conditions and (b) periodic boundary conditions.

Figure 7

Log-log plot of the variance versus the squared mean value of the largest fragment size (charge) distribution. The points are experimental data for events sorted according to the excitation energy [35]. The curves are percolation results for $N=64$ when events are sorted by the bond probability (dashed curve) and by the fraction of open bonds (dotted curve).

Figure 8

Percolation lattice of 64 sites with open boundaries: (a) the normalized variance of $P(s_{\max })$ as a function of the bond probability, (b) the variance versus the squared mean in log-log representation for events binned by p.

Figure 9

The cumulant ratios as functions of the fraction of open bonds; top, the correspondence to the mean value of the bond probability.

Figure 10

The cumulant ratios as functions of the normalized multiplicity.

Figure 11

The cumulant ratios as functions of the normalized total size of complex fragments.

Figure 12

Parameters of the “critical” point as a function of the system size when events are binned according to the multiplicity (left column) and $S_{\mathrm{bound}}$ (right column). The points show percolation values; the curves represent approximations by Eq. (5).