Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On thin local sets of the Gaussian free field

Avelio Sepúlveda

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Abstract

We study how small a local set of the continuum Gaussian free field (GFF) in dimension $d$ has to be to ensure that this set is thin, which loosely speaking means that it captures no GFF mass on itself, in other words, that the field restricted to it is zero. We provide a criterion on the size of the local set for this to happen, and on the other hand, we show that this criterion is sharp by constructing small local sets that are not thin.

Résumé

Nous étudions à quel point un ensemble local du champ libre Gaussien (GFF) en dimension $d$ doit être petit pour être sûr que l’ensemble est fin, ce qui signifie informellement que le GFF ne place pas de masse sur l’ensemble, i.e., que le champ restreint a l’ensemble vaut zéro. Nous donnons des critères portant sur la taille de l’ensemble local qui impliquent cette propriété, et par ailleurs nous montrons que ce critère est optimal en construisant des ensembles locaux petits qui ne sont pas fins.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1797-1813.

Dates
Received: 13 March 2017
Revised: 24 May 2018
Accepted: 22 May 2019
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398886

Digital Object Identifier
doi:10.1214/19-AIHP1005

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Gaussian free field Local sets Thin local sets

Citation

Sepúlveda, Avelio. On thin local sets of the Gaussian free field. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1797--1813. doi:10.1214/19-AIHP1005. https://projecteuclid.org/euclid.aihp/1569398886


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