The Annals of Applied Probability

Nucleation scaling in jigsaw percolation

Janko Gravner and David Sivakoff

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 1 (2017), 395-438.

Dates
Received: November 2015
Revised: April 2016
First available in Project Euclid: 6 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1488790831

Digital Object Identifier
doi:10.1214/16-AAP1206

Mathematical Reviews number (MathSciNet)
MR3619791

Zentralblatt MATH identifier
1362.60086

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Jigsaw percolation nucleation random graph

Citation

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


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