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**1 - 3**of**3**### Given enough choice, simple local rules percolate discontinuously

, 2014

"... Abstract.There is stil much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set ofkrandomly chosen candidate edges at eac ..."

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Abstract.There is stil much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set ofkrandomly chosen candidate edges at each timestep until a giant component emerges. Several Achlioptas processes seem to yield a discontinuous percolation transition, but it was proven by Riordan and Warnke that the transition must be continuous in the thermodynamic limit. However, they also proved that if the numberk(n) of candidate edges increases with the number of nodes, then the percolation transition may be discontinuous. Here we attempt to find the simplest such process which yields a discontinuous transition in the thermodynamic limit. We introduce a process which considers only the degree of candidate edges and not component size. We calculate the critical pointtc=(1−θ(1k))nand rigorously show that the critical window is of sizeO(nk(n)). Ifk(n)grows very slowly, for examplek(n)=logn, the critical window is barely sublinear and hence the phase transition is discontinuous but appears continuous in finite systems. We also present arguments that Achlioptas processes with bounded size rules wil always have continuous percolation transitions even with infinite choice. 1