# How much can the eigenvalues of a random Hermitian matrix fluctuate?

@article{Claeys2021HowMC,
title={How much can the eigenvalues of a random Hermitian matrix fluctuate?},
author={Tom Claeys and Benjamin Fahs and Gaultier Lambert and Christian Webb},
journal={Duke Mathematical Journal},
year={2021}
}
• Published 4 June 2019
• Mathematics
• Duke Mathematical Journal
The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic…
28 Citations

## Figures from this paper

The classical compact groups and Gaussian multiplicative chaos
• Mathematics
Nonlinearity
• 2021
We consider powers of the absolute value of the characteristic polynomial of Haar distributed random orthogonal or symplectic matrices, as well as powers of the exponential of its argument, as a
Liouville quantum gravity from random matrix dynamics
• Mathematics
• 2022
We establish the ﬁrst connection between 2 d Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if ( U t ) is a Brownian motion on the unitary group at
Edge Distribution of Thinned Real Eigenvalues in the Real Ginibre Ensemble
• Mathematics
Annales Henri Poincaré
• 2022
This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed
On the Critical–Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective
• Mathematics
Communications in Mathematical Physics
• 2022
We study the ‘critical moments’ of subcritical Gaussian multiplicative chaos (GMCs) in dimensions $$d \le 2$$ d ≤ 2 . In particular, we establish a fully explicit formula for the leading
Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Mathematics
Nonlinearity
• 2021
We study the orthogonal polynomials and the Hankel determinants associated with the Gaussian weight with two jump discontinuities. When the degree n is finite, the orthogonal polynomials and the
Strong approximation of Gaussian beta-ensemble characteristic polynomials: the hyperbolic regime
• Mathematics
• 2020
We investigate the characteristic polynomials $\varphi_N$ of the Gaussian $\beta$--ensemble for general $\beta>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic)
Optimal Local Law and Central Limit Theorem for $$\beta$$-Ensembles
• Mathematics
Communications in Mathematical Physics
• 2022
In the setting of generic β-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the
Uniform Asymptotics of Toeplitz Determinants with Fisher–Hartwig Singularities
• Benjamin Fahs
• Mathematics
Communications in Mathematical Physics
• 2019
We obtain an asymptotic formula for $$n\times n$$ n × n Toeplitz determinants as $$n\rightarrow \infty$$ n → ∞ , for non-negative symbols with any fixed number of Fisher–Hartwig
Optimal local law and central limit theorem for β-ensembles
In the setting of generic β-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the

## References

SHOWING 1-10 OF 105 REFERENCES
Universality of random matrices and local relaxation flow
• Mathematics
• 2009
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we
Mesoscopic fluctuations for unitary invariant ensembles
Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This
Toeplitz determinants with merging singularities
• Mathematics
• 2015
We study asymptotic behavior for determinants of n×n Toeplitz matrices corresponding to symbols with two Fisher-Hartwig singularities at the distance 2t≥0 from each other on the unit circle. We
On the Characteristic Polynomial¶ of a Random Unitary Matrix
• Mathematics
• 2001
Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at
Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.
• Mathematics
Physical review letters
• 2012
We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the
Extreme gaps between eigenvalues of random matrices
• Mathematics
• 2013
This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian
Convergence of the centered maximum of log-correlated Gaussian fields
• Mathematics
• 2015
We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a
Random matrices with merging singularities and the Painlevé V equation
• Mathematics
• 2016
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form 1 Zn det M 2 tI e nTrV (M) dM, where M is an n n Hermitian matrix, > 1=2