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

TL;DR

This paper analyzes the extreme value statistics of a nonhierarchical, log-correlated Gaussian field, demonstrating that at low temperatures, the Gibbs measure's weights follow a Poisson-Dirichlet distribution, confirming conjectures about their behavior.

Contribution

It proves that the Gibbs weights of the log-correlated Gaussian field converge to a Poisson-Dirichlet distribution, confirming the universality of extreme value statistics in such models.

Findings

Gibbs measure weights converge to Poisson-Dirichlet distribution

Normalized covariance is either 0 or 1 at low temperature

Extremes behave like i.i.d. Gaussian variables and branching Brownian motion

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.