Open Access
February 2017 Nucleation scaling in jigsaw percolation
Janko Gravner, David Sivakoff
Ann. Appl. Probab. 27(1): 395-438 (February 2017). DOI: 10.1214/16-AAP1206


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.


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Janko Gravner. David Sivakoff. "Nucleation scaling in jigsaw percolation." Ann. Appl. Probab. 27 (1) 395 - 438, February 2017.


Received: 1 November 2015; Revised: 1 April 2016; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1362.60086
MathSciNet: MR3619791
Digital Object Identifier: 10.1214/16-AAP1206

Primary: 60K35

Keywords: Jigsaw percolation , nucleation , random graph

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.27 • No. 1 • February 2017
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