Electronic Journal of Probability

Coupling and an application to level-set percolation of the Gaussian free field

Alain-Sol Sznitman

Full-text: Open access

Abstract

In the present article we consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in [9]. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*} $, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 35, 26 pp.

Dates
Received: 24 September 2015
Accepted: 2 March 2016
First available in Project Euclid: 22 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1461332876

Digital Object Identifier
doi:10.1214/16-EJP4563

Mathematical Reviews number (MathSciNet)
MR3492939

Zentralblatt MATH identifier
1336.60194

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82B43: Percolation [See also 60K35]

Keywords
Gaussian free field random interlacements coupling level-set percolation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Sznitman, Alain-Sol. Coupling and an application to level-set percolation of the Gaussian free field. Electron. J. Probab. 21 (2016), paper no. 35, 26 pp. doi:10.1214/16-EJP4563. https://projecteuclid.org/euclid.ejp/1461332876


Export citation

References

  • [1] Athreya, K.B. and Ney, P.E.: Branching Processes. Dover, New York, 2004.
  • [2] Biggins, J.D. and Kyprianou, A.E.: Measure change in multitype branching. Adv. Appl. Probab., 36, (2004), 544–581.
  • [3] Bricmont, J., Lebowitz, J.L. and Maes, C.: Percolation in strongly correlated systems: the massless Gaussian field. J. Stat. Phys., 48, (1987), 1249–1268.
  • [4] Drewitz, A. and Rodriguez, P.-F.: High-dimensional asymptotics for percolation of Gaussian free field level sets. Electron. J. Probab., 20, (2015), 1–39.
  • [5] Eisenbaum, N., Kaspi, H., Marcus, M.B., Rosen, J. and Shi, Z.: A Ray-Knight theorem for symmetric Markov processes. Ann. Probab., 28, (2000), 1781–1796.
  • [6] Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Amer. Math. Soc., 366, (2014), 2089–2119.
  • [7] Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland; Amsterdam, Kodansha, Ltd., Tokyo, 2nd edition, 1989.
  • [8] Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, 1997.
  • [9] Lupu, T.: From loop clusters and random interlacement to the free field. To appear in Ann. Probab., preprint, arXiv:1402.0298.
  • [10] Lupu, T.: Convergence of the two-dimensional random walk loop soup clusters to $CLE$. Preprint, arXiv:1502.06827.
  • [11] Marcus, M.B. and Rosen, J.: Markov processes, Gaussian processes, and local times. Cambridge University Press, 2006.
  • [12] Qian, W. and Werner, W.: Decomposition of Brownian loop-soup clusters. Preprint, arXiv:1509.01180.
  • [13] Pemantle, R.: Automorphism invariant measures on trees. Ann. Probab., 20, (1992), 1549–1566.
  • [14] Rodriguez, P.-F. and Sznitman, A.S.: Phase transition and level-set percolation for the Gaussian free field. Commun. Math. Phys., 320, (2013), 571–601.
  • [15] Reed, M. and Simon, B.: Methods of modern mathematical Physics, volume I. Academic Press, New York, 1972.
  • [16] Rozanov, Yu.A.: Markov Random Fields. Springer, Berlin, 1982.
  • [17] Sabot, C. and Tarrès, P.: Inverting Ray-Knight identity. To appear in Probab. Theor. Relat. Fields, preprint, arXiv:1311.6622.
  • [18] Sznitman, A.S.: An isomorphism theorem for random interlacements. Electron. Commun. Probab., 17, (2012), 1–9.
  • [19] Teixeira, A.: Interlacement percolation on transient weighted graphs. Electron. J. Probab., 14, (2009), 1604–1627.
  • [20] Varopoulos, N.Th.: Long range estimates for Markov chains. Bull. Sc. Math. (2), 109, (1985), 225–252.
  • [21] Zhai, A.: Exponential concentration of cover times. Preprint, arXiv:1407.7617.