Electronic Journal of Probability

Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian free field

Louis-Pierre Arguin and Olivier Zindy

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Abstract

In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions.The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology.This implies Poisson-Dirichlet statistics by general spin glass arguments.In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 59, 19 pp.

Dates
Accepted: 5 June 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067165

Digital Object Identifier
doi:10.1214/EJP.v20-3077

Mathematical Reviews number (MathSciNet)
MR3354619

Zentralblatt MATH identifier
1321.60107

Subjects
Primary: 60G15: Gaussian processes 60F05: Central limit and other weak theorems
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60G70: Extreme value theory; extremal processes 82B26: Phase transitions (general)

Keywords
Gaussian free field Gibbs measure Poisson-Dirichlet variable spin glasses

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Arguin, Louis-Pierre; Zindy, Olivier. Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian free field. Electron. J. Probab. 20 (2015), paper no. 59, 19 pp. doi:10.1214/EJP.v20-3077. https://projecteuclid.org/euclid.ejp/1465067165


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