Bohman-Frieze-Wormald model on the lattice, yielding a discontinuous percolation transition

TL;DR

This study investigates the Bohman-Frieze-Wormald (BFW) percolation model on lattices, revealing strongly discontinuous phase transitions in two and three dimensions, with detailed analysis of cluster structures and size distributions across dimensions.

Contribution

It provides the first numerical evidence of strongly discontinuous percolation transitions of the BFW model on square and cubic lattices, including analysis of cluster properties and the tree-like version across multiple dimensions.

Findings

Discontinuous transition observed in 2D and 3D lattices.

Clusters at the threshold are compact with fractal surfaces.

Discontinuous transitions persist in the tree-like version across dimensions.

Abstract

The BFW model introduced by Bohman, Frieze, and Wormald [Random Struct. Algorithms, 25, 432 (2004)] and recently investigated in the framework of discontinuous percolation by Chen and D'Souza [Phys. Rev. Lett., 106, 115701 (2011)], is studied on the square and simple-cubic lattices. In two and three dimensions, we find numerical evidence for a strongly discontinuous transition. In two dimensions, the clusters at the threshold are compact with a fractal surface of fractal dimension $d_f=1.49\pm0.02$. On the simple-cubic lattice, distinct jumps in the size of the largest cluster are observed. We proceed to analyze the tree-like version of the model, where only merging bonds are sampled, for dimension two to seven. The transition is again discontinuous in any considered dimension. Finally, the dependence of the cluster-size distribution at the threshold on the spatial dimension is also investigated.