Gaussian model of explosive percolation in three and higher dimensions

TL;DR

This paper investigates the Gaussian model of discontinuous percolation in three and higher dimensions, revealing a consistent discontinuous transition characterized by a finite jump in the order parameter and fractal external perimeters.

Contribution

It extends the Gaussian percolation model to higher dimensions, confirming the discontinuous nature and analyzing key properties across dimensions up to six and the mean-field limit.

Findings

Discontinuous transition with finite order parameter jump in all dimensions

Fractal external perimeter with dimension ~2.5 across dimensions

Lower bound established for percolation threshold in finite-cluster models

Abstract

The Gaussian model of discontinuous percolation, recently introduced by Ara\'ujo 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.