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David Sivakoff : Nucleation scaling in jigsaw percolation

Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdos-Renyi graph with edge probability p, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition {V}. In some generality, for puzzle graphs with N vertices of degrees about D, this probability is close to 1 or 0 depending on whether pD(log N) is large or small. We give more detailed results for the one dimensional cycle and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.

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