The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 27, Number 1 (2017), 395-438.
Nucleation scaling in jigsaw percolation
Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic “puzzle graph” by using connectivity properties of a random “people graph” on the same set of vertices. We presume the Erdős–Rényi 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 $pD\log N$ 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. The paper strengthens several results of Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.
Ann. Appl. Probab., Volume 27, Number 1 (2017), 395-438.
Received: November 2015
Revised: April 2016
First available in Project Euclid: 6 March 2017
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Gravner, Janko; Sivakoff, David. Nucleation scaling in jigsaw percolation. Ann. Appl. Probab. 27 (2017), no. 1, 395--438. doi:10.1214/16-AAP1206. https://projecteuclid.org/euclid.aoap/1488790831