Zusammenfassung
The Gaussian model of discontinuous percolation, recently introduced by
Araujo and Herrmann Phys. Rev. Lett. 105, 035701 (2010), is
numerically investigated in three dimensions, disclosing a discontinuous
transition. For the simple cubic lattice, in the thermodynamic limit we report a finite jump of the order parameter J = 0.415 +/- 0.005. The
largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension d(A) = 2.5 +/- 0.2. The study is
extended to hypercubic lattices up to six dimensions and to the
mean-field limit (infinite dimension). We find that, in all considered
dimensions, the percolation transition is discontinuous. The value of
the jump in the order parameter, the maximum of the second moment, and
the percolation threshold are analyzed, revealing interesting features
of the transition and corroborating its discontinuous nature in all
considered dimensions. We also show that the fractal dimension of the
external perimeter, for any dimension, is consistent with the one from
bridge percolation and establish a lower bound for the percolation
threshold of discontinuous models with a finite number of clusters at
the threshold.
Nutzer