The Annals of Applied Probability

Poisson–Dirichlet statistics for the extremes of a log-correlated Gaussian field

Louis-Pierre Arguin and Olivier Zindy

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Abstract

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval $[0,1]$, and its correlation structure is nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm. Math. Phys. 236 (2003) 449–475] (see also Barral and Mandelbrot [Probab. Theory Related Fields 124 (2002) 409–430]), and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001) 026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008) 372001]. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either $0$ or $1$. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson–Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The method of proof is robust and is adaptable to other log-correlated Gaussian fields.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 4 (2014), 1446-1481.

Dates
First available in Project Euclid: 14 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1400073654

Digital Object Identifier
doi:10.1214/13-AAP952

Mathematical Reviews number (MathSciNet)
MR3211001

Zentralblatt MATH identifier
1301.60042

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
Log-correlated Gaussian fields Gibbs measure Poisson–Dirichlet variable tree approximation spin glasses

Citation

Arguin, Louis-Pierre; Zindy, Olivier. Poisson–Dirichlet statistics for the extremes of a log-correlated Gaussian field. Ann. Appl. Probab. 24 (2014), no. 4, 1446--1481. doi:10.1214/13-AAP952. https://projecteuclid.org/euclid.aoap/1400073654


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