Electronic Communications in Probability

The incipient giant component in bond percolation on general finite weighted graphs

David Aldous

Full-text: Open access

Abstract

On a large finite connected graph let edges $e$ become “open” at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean.

Article information

Source
Electron. Commun. Probab. Volume 21 (2016), paper no. 68, 9 pp.

Dates
Received: 26 April 2016
Accepted: 7 September 2016
First available in Project Euclid: 21 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1474462208

Digital Object Identifier
doi:10.1214/16-ECP21

Zentralblatt MATH identifier
1348.60136

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

Keywords
bond percolation incipient giant component concentration inequalities

Rights
Creative Commons Attribution 4.0 International License.

Citation

Aldous, David. The incipient giant component in bond percolation on general finite weighted graphs. Electron. Commun. Probab. 21 (2016), paper no. 68, 9 pp. doi:10.1214/16-ECP21. https://projecteuclid.org/euclid.ecp/1474462208.


Export citation

References

  • [1] David Aldous, Interacting particle systems as stochastic social dynamics, Bernoulli 19 (2013), no. 4, 1122–1149.
  • [2] D.J. Aldous, Weak concentration for first passage percolation times on graphs and general increasing set-valued processes, arXiv:1604.06418, 2016.
  • [3] D.J. Aldous and Xiang Li, A framework for imperfectly observed networks, 2016, In Preparation.
  • [4] Noga Alon, Itai Benjamini, and Alan Stacey, Percolation on finite graphs and isoperimetric inequalities, Ann. Probab. 32 (2004), no. 3A, 1727–1745.
  • [5] Noga Alon and Joel H. Spencer, The probabilistic method, third ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008, With an appendix on the life and work of Paul Erdős.
  • [6] Antonio Auffinger, Jack Hanson, and Michael Damron, 50 years of first passage percolation, 2015, arXiv:1511.03262.
  • [7] Itai Benjamini, Stéphane Boucheron, Gábor Lugosi, and Raphaël Rossignol, Sharp threshold for percolation on expanders, Ann. Probab. 40 (2012), no. 1, 130–145.
  • [8] Shankar Bhamidi, Remco van der Hofstad, and Gerard Hooghiemstra, First passage percolation on random graphs with finite mean degrees, Ann. Appl. Probab. 20 (2010), no. 5, 1907–1965.
  • [9] Shankar Bhamidi, Remco van der Hofstad, and Gerard Hooghiemstra, First passage percolation on the Erdős-Rényi random graph, Combin. Probab. Comput. 20 (2011), no. 5, 683–707.
  • [10] Béla Bollobás and Oliver Riordan, The phase transition in the Erdős-Rényi random graph process, Erdös centennial, Bolyai Soc. Math. Stud., vol. 25, János Bolyai Math. Soc., Budapest, 2013, pp. 59–110.
  • [11] C. Borgs, J. T. Chayes, H. Kesten, and J. Spencer, The birth of the infinite cluster: finite-size scaling in percolation, Comm. Math. Phys. 224 (2001), no. 1, 153–204, Dedicated to Joel L. Lebowitz.
  • [12] Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, Gordon Slade, and Joel Spencer, Random subgraphs of finite graphs. I. The scaling window under the triangle condition, Random Structures Algorithms 27 (2005), no. 2, 137–184.
  • [13] Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, Gordon Slade, and Joel Spencer, Random subgraphs of finite graphs. II. The lace expansion and the triangle condition, Ann. Probab. 33 (2005), no. 5, 1886–1944.
  • [14] Fan Chung, Paul Horn, and Linyuan Lu, Percolation in general graphs, Internet Math. 6 (2009), no. 3, 331–347 (2010).
  • [15] Olle Häggström and Johan Jonasson, Uniqueness and non-uniqueness in percolation theory, Probab. Surv. 3 (2006), 289–344.
  • [16] Harry Kesten, On the speed of convergence in first-passage percolation, Ann. Appl. Probab. 3 (1993), no. 2, 296–338.
  • [17] Harry Kesten, First-passage percolation, From Classical to Modern Probability, Progr. Probab., vol. 54, Birkhäuser, Basel, 2003, pp. 93–143.
  • [18] Asaf Nachmias and Yuval Peres, Critical percolation on random regular graphs, Random Structures Algorithms 36 (2010), no. 2, 111–148.