## The Annals of Applied Probability

### Nucleation scaling in jigsaw percolation

#### 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
Revised: April 2016
First available in Project Euclid: 6 March 2017

https://projecteuclid.org/euclid.aoap/1488790831

Digital Object Identifier
doi:10.1214/16-AAP1206

Mathematical Reviews number (MathSciNet)
MR3619791

Zentralblatt MATH identifier
1362.60086

#### 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

#### References

• [1] Aizenman, M. and Lebowitz, J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A 21 3801–3813.
• [2] Balogh, J., Bollobás, B., Duminil-Copin, H. and Morris, R. (2012). The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc. 364 2667–2701.
• [3] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
• [4] Bollobás, B., Kohayakawa, Y. and Łuczak, T. (1994). On the evolution of random Boolean functions. In Extremal Problems for Finite Sets (P. Frankl, Z. Füredi, G. Katona and D. Miklós, eds.). Bolyai Society Mathematical Studies 3. János Bolyai Mathematical Society, Budapest.
• [5] Bollobás, B., Riordan, O., Slivken, E. and Smith, P. (2015). The threshold for jigsaw percolation on random graphs. Preprint. Available at arXiv:1503.05186.
• [6] Brummitt, C. D., Chatterjee, S., Dey, P. S. and Sivakoff, D. (2015). Jigsaw percolation: What social networks can collaboratively solve a puzzle? Ann. Appl. Probab. 25 2013–2038.
• [7] Finch, S. R. (1999). Several constants arising in statistical mechanics. Ann. Comb. 3 323–335.
• [8] Fisch, R., Gravner, J. and Griffeath, D. (1993). Metastability in the Greenberg–Hastings model. Ann. Appl. Probab. 3 935–967.
• [9] Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 2993–3002.
• [10] Gravner, J. (1996). Percolation times in two-dimensional models for excitable media. Electron. J. Probab. 1 19 pp. (electronic).
• [11] Gravner, J. and Holroyd, A. E. (2009). Local bootstrap percolation. Electron. J. Probab. 14 385–399.
• [12] Gravner, J., Holroyd, A. E. and Morris, R. (2012). A sharper threshold for bootstrap percolation in two dimensions. Probab. Theory Related Fields 153 1–23.
• [13] Holroyd, A. E. (2003). Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125 195–224.
• [14] Kesten, H. (1982). Percolation Theory for Mathematicians. Progress in Probability and Statistics 2. Birkhäuser, Boston, MA.
• [15] Kesten, H. (1990). Asymptotics in high dimensions for percolation. In Disorder in Physical Systems (G. Grimmett and D. J. A. Welsh, eds.). Oxford Sci. Publ. 219–240. Oxford Univ. Press, New York.
• [16] O’Connell, N. (1998). Some large deviation results for sparse random graphs. Probab. Theory Related Fields 110 277–285.
• [17] Russo, L. (1981). On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229–237.
• [18] Sedgewick, R. (1997). Algorithms in C, 3rd ed. Addison-Wesley, Reading, MA.
• [19] Sivakoff, D. (2014). Site percolation on the $d$-dimensional Hamming torus. Combin. Probab. Comput. 23 290–315.
• [20] Wierman, J. C. (1995). Substitution method critical probability bounds for the square lattice site percolation model. Combin. Probab. Comput. 4 181–188.