Nucleation scaling in jigsaw percolation

Abstract

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic \puzzle by using connectivity properties of a random \people on the same set of vertices. We presume the Erd} os-R enyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD logN is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. Main tools come from analysis of bootstrap percolation and other local nucleation-and-growth models. The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and Sivako who