## Electronic Journal of Probability

### Bootstrap percolation on products of cycles and complete graphs

#### Abstract

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $\theta$ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 29, 20 pp.

Dates
Accepted: 1 March 2017
First available in Project Euclid: 24 March 2017

https://projecteuclid.org/euclid.ejp/1490320845

Digital Object Identifier
doi:10.1214/17-EJP43

Mathematical Reviews number (MathSciNet)
MR3629873

Zentralblatt MATH identifier
1361.60092

#### Citation

Gravner, Janko; Sivakoff, David. Bootstrap percolation on products of cycles and complete graphs. Electron. J. Probab. 22 (2017), paper no. 29, 20 pp. doi:10.1214/17-EJP43. https://projecteuclid.org/euclid.ejp/1490320845

#### References

• [AdL] J. Adler and U. Lev, Bootstrap percolation: visualizations and applications. Brazilian Journal of Physics 33 (2003), 641–644.
• [AiL] M. Aizenman and J. L. Lebowitz, Metastability effects in bootstrap percolation. Journal of Physics A 21 (1988), 3801–3813.
• [AGG] R. Arratia, L. Goldstein, L. Gordon, Two Moments Suffice for Poisson Approximations: The Chen-Stein Method., Annals of Probability 17 (1989), 9–25.
• [BBLN] P. Balister, B. Bollobás, J. Lee and B. Narayanan, Line percolation. arXiv:1403.6851
• [BB1] J. Balogh and B. Bollobás, Sharp thresholds in bootstrap percolation., Physica A 326 (2003), 305–312.
• [BB2] J. Balogh and B. Bollobás, Bootstrap percolation on the hypercube. Probability Theory and Related Fields 134 (2006), 624–648.
• [BBDM] J. Balogh, B. Bollobás, H. Duminil-Copin, R. Morris, The sharp threshold for bootstrap percolation in all dimensions. Transactions of the American Mathematical Society 364 (2012), pp. 2667–2701.
• [BBM] J. Balogh, B. Bollobás, R. Morris, Bootstrap percolation in three dimensions. Annals of Probability 37 (2009), 1329–1380.
• [BHJ] A. D. Barbour, L. Holst, S. Janson, Poisson approximation. Oxford University Press (1992).
• [CLR] J. Chalupa, P. L. Leath, G. R. Reich, Bootstrap percolation on a Bethe lattice. Journal of Physics C 12 (1979), L31–L35.
• [DE] H. Duminil-Copin, A. C. D. van Enter, Sharp metastability threshold for an anisotropic bootstrap percolation model. Annals of Probability 41 (2013), 1218–1242.
• [FK] E. Friedgut and G. Kalai, Every monotone graph property has a sharp threshold, Proceedings of the American Mathematical Society 10 (1996), 2993–3002.
• [Gar] D. Gardy, Some results on the asymptotic behaviour of coefficients of large powers of functions. Discrete Mathematics 139 (1995) 189–217.
• [GG] J. Gravner, D. Griffeath, First passage times for threshold growth dynamics on ${\mathbb Z}^2$. Annals of Probability 24 (1996), 1752–1778.
• [GHPS] J. Gravner, C. Hoffman, J. Pfeiffer, D. Sivakoff, Bootstrap percolation on the Hamming torus. Annals of Applied Probability 25 (2015), 287–323.
• [GHM] J. Gravner, A. E. Holroyd, and R. Morris, A sharper threshold for bootstrap percolation in two dimensions. Probability Theory and Related Fields 153 (2012), 1–23.
• [Hol1] A. E. Holroyd, Sharp metastability threshold for two-dimensional bootstrap percolation. Probability Theory and Related Fields 125 (2003), 195–224.
• [Hol2] A. E. Holroyd, Astonishing cellular automata. Bulletin du Centre de Recherches Mathematiques 13 (2007), 10–13.
• [HLR] A. E. Holroyd, T. M. Liggett, and D. Romik, Integrals, partitions, and cellular automata. Transactions of the American Mathematical Society 356 (2004), pp. 3349–3368.
• [LZ] S. P. Lalley and X. Zheng, Spatial epidemics and local times for critical branching random walks in dimensions $2$ and $3$. Probability Theory and Related Fields 148(3) (2010), 527–566.
• [Sch] R. H. Schonmann, On the behavior of some cellular automata related to bootstrap percolation, Annals of Probability 20 (1992), 174–193.
• [Siv] D. Sivakoff, Site percolation on the $d$-dimensional Hamming torus. Combinatorics, Probability and Computing 23 (2014), 290–315.
• [Sli] E. Slivken, Bootstrap percolation on the Hamming torus with threshold 2., arXiv:1407.2317
• [TV] T. Turova and T. Vallier, Bootstrap percolation on a graph with random and local connections., Journal of Statistical Physics 160 (2015), 1249–1276.
• [vEn] A. C. D. van Enter, Proof of Straley’s argument for bootstrap percolation. Journal of Statistical Physics 48 (1987), 943–945.