Electronic Journal of Probability

Metastability for the contact process on the configuration model with infinite mean degree

Van Hao Can and Bruno Schapira

Full-text: Open access

Abstract

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two.

We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward

an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained previously.  

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 26, 22 pp.

Dates
Accepted: 9 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067132

Digital Object Identifier
doi:10.1214/EJP.v20-3859

Mathematical Reviews number (MathSciNet)
MR3325096

Zentralblatt MATH identifier
1327.82051

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Keywords
Contact process random graphs configuration model metastability

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Can, Van Hao; Schapira, Bruno. Metastability for the contact process on the configuration model with infinite mean degree. Electron. J. Probab. 20 (2015), paper no. 26, 22 pp. doi:10.1214/EJP.v20-3859. https://projecteuclid.org/euclid.ejp/1465067132


Export citation

References

  • Berger, Noam; Borgs, Christian; Chayes, Jennifer T.; Saberi, Amin. On the spread of viruses on the internet. Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 301–310, ACM, New York, 2005.
  • Benjamini, Itai; Schramm, Oded. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), no. 23, 13 pp. (electronic).
  • Chatterjee, Shirshendu; Durrett, Rick. Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37 (2009), no. 6, 2332–2356.
  • Cranston, Michael; Mountford, Thomas; Mourrat, Jean-Christophe; Valesin, Daniel. The contact process on finite homogeneous trees revisited. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 385–408.
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. xii+332 pp. ISBN: 3-540-65995-1
  • Mountford, Thomas; Valesin, Daniel; Yao, Qiang. Metastable densities for the contact process on power law random graphs. Electron. J. Probab. 18 (2013), No. 103, 36 pp.
  • T. Mountford, J.-C. Mourrat, D. Valesin, Q. Yao. Exponential extinction time of the contact process on finite graphs, arXiv:1203.2972.
  • Mountford, T. S. A metastable result for the finite multidimensional contact process. Canad. Math. Bull. 36 (1993), no. 2, 216–226.
  • J.-C. Mourrat, D. Valesin. Phase transition of the contact process on random regular graphs, arXiv:1405.0865.
  • Pemantle, Robin. The contact process on trees. Ann. Probab. 20 (1992), no. 4, 2089–2116.
  • van den Esker, Henri; van der Hofstad, Remco; Hooghiemstra, Gerard; Znamenski, Dmitri. Distances in random graphs with infinite mean degrees. Extremes 8 (2005), no. 3, 111–141 (2006).
  • R. Van der Hofstad Random graphs and complex networks. Available at http://www.win.tue.nl/~rhofstad/NotesRGCN.html.