Gaussian model of explosive percolation in three and higher dimensions

Abstract

The Gaussian model of discontinuous percolation, recently introduced by Araújo 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, $J=0.415\pm 0.005$. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension $d_A=2.5\pm 0.2$. 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 finite number of clusters at the threshold.

References in corpus (15)

• Self-organized adaptation of a simple neural circuit enables complex robot behaviour
• Impact of Single Links in Competitive Percolation -- How complex networks grow under competition
• Explosive Percolation is Continuous, but with Unusual Finite Size Behavior
• Explosive percolation in scale-free networks
• Explosive percolation via control of the largest cluster
• Strongly discontinuous explosive percolation with multiple giant components
• Tricritical point in explosive percolation
• Continuity of the Explosive Percolation Transition
• Using explosive percolation in analysis of real-world networks
• Optimal path cracks in correlated and uncorrelated lattices
• New efficient methods to calculate watersheds
• Impact of Perturbations on Watersheds
• Spreading with immunization in high dimensions
• Criterion for explosive percolation transitions on complex networks
• Explosive percolation in graphs

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