Maximum of a log-correlated Gaussian field

By Thomas Madaule. 2013

83 citations recorded by the source corpus.

We study the maximum of a Gaussian field on $[0,1]^\d$ ($\d \geq 1$) whose correlations decay logarithmically with the distance. Kahane \cite{Kah85} introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas \cite{DRSV12a} \cite{DRSV12b} extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in \cite{DRSV12a} several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in \cite{DRSV12a}: we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8973a5f26240f203ac0a6e0c9802327af95d640c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Zhan Shi
2. Joseph Najnudel
3. Ofer Zeitouni
4. Jian Ding
5. L. Arguin
6. Rishideep Roy
7. David Belius
8. P. Bourgade

In Mathematical Physics

1. Joseph Najnudel
2. L. Arguin
3. David Belius
4. Zhan Shi
5. Ofer Zeitouni
6. Jian Ding
7. P. Bourgade
8. Rishideep Roy

Citations in the corpus

• Maximum of the Characteristic Polynomial of Random Unitary Matrices (2015, background): “In particular, convergence of the maximum of a related model of log-correlated Gaussian field was proved in [46].”
• Branching random walks (2015, result): “By comparison to analoguous results for branching random walks, many deep results have been recently established for Gaussian free fields and more general logarithmically correlated Gaussian fields (Bolthausen, Deuschel and Giacomin [61], Madaule [183], Biskup and Louidor [60], Ding, Roy and Zeitoun”
• On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, methodology): “In the case of log-correlated Gaussian fields on [0, 1]d (d ≥ 1) a limit theorem has been proven by Madaule [24], and then generalized by Ding, Roy and Zeitouni [14].”
• Extrema of log-correlated random variables: Principles and Examples (2016, background): “[23] (see Biskup & Louidor [17] for the expression in terms of the derivative martingale), for a type of Gaussian Multiplicative Chaos by Madaule [50], and finally in great generality for log-correlated Gaussian fields by Ding et al.”
• Convergence of the centered maximum of log-correlated Gaussian fields (2015, background): “Of greatest relevance to the current paper are [1, 6, 7, 17, 19].”
• Maxima of a randomized Riemann zeta function, and branching random walks (2015, background): “It has since been extended to more general branching random walks by several authors, for example, [1, 2, 11], and to log-correlated Gaussian fields see, for example, [12, 23].”