PerspectiveMathematics

Getting the Jump on Explosive Percolation

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Science  08 Mar 2013:
Vol. 339, Issue 6124, pp. 1159-1160
DOI: 10.1126/science.1235032

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Summary

Percolation refers to the formation of long-range connectedness or conductivity in random systems. Simple models for percolation were independently devised in the areas of polymer science (1) and mathematics (2) in the 1940s and '50s, and have been both a persistent theoretical challenge and an enduring practical paradigm ever since. In the past decade, percolation has become a central problem in probability theory, and has figured in the work of two recent Fields medalists (3). A recent and somewhat controversial development concerns looking at the dynamics of percolation under various global bond selection rules and how percolating systems make the transition from being disconnected (or comprising a group of disconnected clusters) to being fully connected. It had been shown that the transition can proceed explosively, in which the transition is discontinuous (4), but that scenario was later challenged when it was shown that for some specific systems, such a transition is in fact continuous. On page 1185 of this issue, Cho et al. (5) show analytically and numerically that the explosive percolation transition can be either continuous or discontinuous, depending on the bias against certain "bridging" bonds and the dimensionality of the system.