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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