The Annals of Probability

Sharp threshold for percolation on expanders

Itai Benjamini, Stéphane Boucheron, Gábor Lugosi, and Raphaël Rossignol

Full-text: Open access

Abstract

We study the appearance of the giant component in random subgraphs of a given large finite graph G = (V, E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then for any c ∈ ]0, 1[, the property that the random subgraph contains a giant component of size c|V| has a sharp threshold.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 130-145.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325604999

Digital Object Identifier
doi:10.1214/10-AOP610

Mathematical Reviews number (MathSciNet)
MR2917769

Zentralblatt MATH identifier
1239.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Keywords
Percolation random graph expander giant component sharp threshold

Citation

Benjamini, Itai; Boucheron, Stéphane; Lugosi, Gábor; Rossignol, Raphaël. Sharp threshold for percolation on expanders. Ann. Probab. 40 (2012), no. 1, 130--145. doi:10.1214/10-AOP610. https://projecteuclid.org/euclid.aop/1325604999


Export citation

References

  • Alon, N., Benjamini, I. and Stacey, A. (2004). Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 1727–1745.
  • Benjamini, I., Nachmias, A. and Peres, Y. (2009). Is the critical percolation probability local? Probab. Theory Related Fields 149 261–269.
  • Benjamini, I. and Rossignol, R. (2008). Submean variance bound for effective resistance of random electric networks. Comm. Math. Phys. 280 445–462.
  • Benjamini, I. and Schramm, O. (1996). Percolation beyond ℤd, many questions and a few answers. Electron. Comm. Probab. 1 71–82.
  • Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596.
  • Falik, D. and Samorodnitsky, A. (2007). Edge-isoperimetric inequalities and influences. Combin. Probab. Comput. 16 693–712. Available from http://arxiv.org/pdf/math.CO/0512636.
  • Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge Univ. Press, Cambridge.
  • Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 2993–3002.
  • Janson, S. (2004). Large deviations for sums of partly dependent random variables. Random Structures Algorithms 24 234–248.
  • Lyons, R. and Peres, Y. (2011). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge. To appear. Available at http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.
  • Nachmias, A. and Peres, Y. (2010). Critical percolation on random regular graphs. Random Structures Algorithms 36 111–148.
  • Pittel, B. (2008). Edge percolation on a random regular graph of low degree. Ann. Probab. 36 1359–1389.
  • Rossignol, R. (2006). Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab. 34 1707–1725.