Maximum of the Characteristic Polynomial of Random Unitary Matrices

By L. Arguin, David Belius, P. Bourgade. Communications in Mathematical Physics, 2015

127 citations recorded by the source corpus.

It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a N×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N\times N}$$\end{document} random unitary matrix sampled from the Haar measure grows like CN/(logN)3/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${CN/({\rm log} N)^{3/4}}$$\end{document} for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range [N1-ε,N1+ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[N^{1 - \varepsilon},N^{1 + \varepsilon}]}$$\end{document}, for arbitrarily small ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon}$$\end{document}. The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/f-noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/50fdfff26e39d66108f32bf2d08bba9a642c3fcc

How researchers see this paper

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

In Geometry and Topology

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

The maximum of the CUE field (2016, methodology): “The attempt to use techniques developed in the context of extrema of logarithmically correlated fields, and in particular a modified version of the second moment method, to the study of U∗ is natural and in fact is behind the study [ABB15].”
Extrema of log-correlated random variables: Principles and Examples (2016, result): “˙2 is already consistent with this result. Kistler’s multiscale renement of the second moment method as described in Section 3 can be adapted to prove the leading order of the conjecture. Theorem 7 ([6]). For N 2N, let UN be a random matrix sampled uniformly from the group of N N unitary matrices. W”
On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, methodology): “1 is the correct one, which correspond to the precision of the result by Arguin, Belius and Bourgade [3] for random matrices.”
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, methodology): “The identification of an approximate branching random walk structure within the sum ∑ p≤X p −s was used in [3] to study the extrema of a random model of the zeta function, and in the subsequent works regarding the large values of characteristic polynomials [2,12,26,33] and of the zeta function [32].”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, result): “ type of conjectures have been recently obtained for the random truncated model of the logarithm of the Riemann zeta function and the logarithm of the characteristic polynomial of CUE random matrices [1, 2, 59, 19], where very precise asymptotics for the maximum of the relevant log-correlated ﬁelds ”