Ordinary Percolation with Discontinuous Transitions

S. Boettcher; V. Singh; R. M. Ziff

doi:10.1038/ncomms1774

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arXiv · 1110.4288

Ordinary Percolation with Discontinuous Transitions

S. Boettcher·V. Singh·R. M. Ziff

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Abstract

Percolation on a one-dimensional lattice and fractals such as the Sierpinski gasket is typically considered to be trivial because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a nontrivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here, we provide a simple example of this transition in form of a small-world network consisting of a one-dimensional lattice combined with a hierarchy of long-range bonds that reveals many features of the transition in a mathematically rigorous manner.

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S. Boettcher, V. Singh, R. M. Ziff. 2011-10-19. Ordinary Percolation with Discontinuous Transitions. https://doi.org/10.1038/ncomms1774

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