Extrema of log-correlated random variables: Principles and Examples

By L. Arguin. 2016

48 citations recorded by the source corpus.

These notes were written for the mini-course "Extrema of log-correlated random variables: Principles and Examples" at the Introductory School held in January 2015 at the Centre International de Rencontres Math\'ematiques in Marseille. There have been many advances in the understanding of the high values of log-correlated random fields from the physics and mathematics perspectives in recent years. These fields admit correlations that decay approximately like the logarithm of the inverse of the distance between index points. Examples include branching random walks and the two-dimensional Gaussian free field. In this paper, we review the properties of such fields and survey the progress in describing the statistics of their extremes. The branching random walk is used as a guiding example to prove the correct leading and subleading order of the maximum following the multiscale refinement of the second moment method of Kistler. The approach sheds light on a conjecture of Fyodorov, Hiary & Keating on the maximum of the Riemann zeta function on an interval of the critical line and of the characteristic polynomial of random unitary matrices.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e7de99ff48e8f4b2873b55313c587181a9b8a939

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In Finance

1. David Belius
2. L. Arguin
3. P. Bourgade
4. Kannan Soundararajan
5. Maksym Radziwill
6. Zhan Shi

In Mathematical Physics

1. L. Arguin
2. P. Bourgade
3. Maksym Radziwill
4. Kannan Soundararajan
5. David Belius
6. Joseph Najnudel
7. Paul D. Nelson
8. Ofer Zeitouni

Citations in the corpus

• Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, methodology): “of random walks, Gaussian multiplicative chaos, random matrices and the Riemann zeta function. We do not give here a list of the many rigorous works on this topic in recent years, pointing instead to [1] and the references therein. 1.3. About the proof. The short proof of the upper bound in Theorem ”