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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 $h_{*}>0$.


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Level-set percolation Gaussian free field Transient trees Random interlacements


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.

Export citation


  • [1] R. Abraham and J.-F. Delmas. An introduction to Galton–Watson trees and their local limits, 2015. Available at
  • [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.
  • [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
  • [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.
  • [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
  • [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.
  • [15] C. Sabot and P. Tarrès. Inverting Ray–Knight identity. Probab. Theory Relat. Fields. To appear. Available at 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.