Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Level-set percolation for the Gaussian free field on a transient tree

Angelo Abächerli and Alain-Sol Sznitman

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Abstract

We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton–Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_{*}$ and $u_{*}$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_{*}<\sqrt{2u}_{*}$ in a broad enough set-up, but provide an example where $0=h_{*}=u_{*}$ occurs. We also obtain some sufficient conditions ensuring that $h_{*}>0$.

Résumé

Nous étudions la percolation de niveau pour le champ libre gaussien sur des arbres transients, par exemple sur des arbres de Galton–Watson surcritiques conditionnés à survivre. Des théorèmes de type isomorphisme de Dynkin récemment obtenus offrent un outil de comparaison avec la percolation de l’ensemble vacant pour les entrelacs aléatoires, qui se trouve être plus simple à étudier dans le cas des arbres. Si $h_{*}$ et $u_{*}$ désignent les valeurs critiques respectives de la percolation de niveau du champ libre gaussien, et de l’ensemble vacant des entrelacs aléatoires, nous montrons dans un cadre assez général que $h_{*}<\sqrt{2u}_{*}$, mais présentons un exemple pour lequel on a les égalités $0=h_{*}=u_{*}$. Nous obtenons aussi des conditions suffisantes qui impliquent que $h_{*}>0$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 173-201.

Dates
Received: 24 June 2016
Revised: 20 September 2016
Accepted: 27 September 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030824

Digital Object Identifier
doi:10.1214/16-AIHP799

Mathematical Reviews number (MathSciNet)
MR3765885

Zentralblatt MATH identifier
06880050

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82B43: Percolation [See also 60K35]

Keywords
Level-set percolation Gaussian free field Transient trees Random interlacements

Citation

Abächerli, Angelo; Sznitman, Alain-Sol. Level-set percolation for the Gaussian free field on a transient tree. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 173--201. doi:10.1214/16-AIHP799. https://projecteuclid.org/euclid.aihp/1519030824


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References

  • [1] R. Abraham and J.-F. Delmas. An introduction to Galton–Watson trees and their local limits, 2015. Available at https://hal.archives-ouvertes.fr/hal-01164661.
  • [2] J. Bricmont, J. L. Lebowitz and C. Maes. Percolation in strongly correlated systems: The massless Gaussian field. J. Stat. Phys. 48 (5/6) (1987) 1249–1268.
  • [3] J. Černý and A. Teixeira. From random walk trajectories to random interlacements. Ensaios Matemáticos 23 2012.
  • [4] A. Drewitz, B. Ráth and A. Sapozhnikov. An Introduction to Random Interlacements. In SpringerBriefs in Mathematics. Berlin, 2014.
  • [5] A. Drewitz and P.-F. Rodriguez. High-dimensional asymptotics for percolation of Gaussian free field level sets. Electron. J. Probab. 20 (47) (2015) 1–39.
    Mathematical Reviews (MathSciNet): MR3339867
  • [6] M. Folz. Volume growth and stochastic completeness of graphs. Trans. Amer. Math. Soc. 366 (4) (2014) 2089–2119.
  • [7] G. Giacomin. Aspects of Statistical Mechanics of Random Surfaces. Notes of Lectures at IHP (Fall 2001), Version of February 24, 2003. Available at http://www.proba.jussieu.fr/pageperso/giacomin/pub/IHP.ps.
  • [8] G. Grimmett and H. Kesten. Random electrical networks on complete graphs II: Proofs. Available at arXiv:math/0107068.
  • [9] T. Lupu. From loop clusters and random interlacement to the free field. Ann. Probab. To appear. Available at arXiv:1402.0298.
    arXiv: 1402.0298
  • [10] R. Lyons. Random walks and percolation on trees. Ann. Probab. 18 (3) (1990) 931–958.
  • [11] R. Lyons. Random walks, capacity and percolation on trees. Ann. Probab. 20 (4) (1992) 2043–2088.
  • [12] R. Lyons and Y. Peres. Probability on Trees and Networks, 2016. Available at http://pages.iu.edu/~rdlyons/.
  • [13] P.-F. Rodriguez and A. S. Sznitman. Phase transition and level-set percolation for the Gaussian free field. Comm. Math. Phys. 320 (2013) 571–601.
  • [14] J. Rosen. Lectures on Isomorphism Theorems. Preprint. Available at arXiv:1407.1559.
    arXiv: 1407.1559
  • [15] C. Sabot and P. Tarrès. Inverting Ray–Knight identity. Probab. Theory Relat. Fields. To appear. Available at arXiv:1311.6622.
    arXiv: 1311.6622
  • [16] A. S. Sznitman. An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17 (9) (2012) 1–9.
  • [17] A. S. Sznitman. Coupling and an application to level-set percolation of the Gaussian free field. Electron. J. Probab. 21 (35) (2016) 1–26.
  • [18] A. S. Sznitman. Topics in Occupation Times and Gaussian Free Fields. Zurich Lectures in Advanced Mathematics. EMS, Zurich, 2012.
  • [19] M. Tassy. Random interlacements on Galton-Watson trees. Electron. Commun. Probab. 15 (2010) 562–571.
  • [20] A. Teixeira. Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 (2009) 1604–1627.
  • [21] A. Zhai. Exponential concentration of cover times. Preprint. Available at arXiv:1407.7617.
    arXiv: 1407.7617

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