Maximum of the characteristic polynomial of the Ginibre ensemble

Lambert, Gaultier

doi:10.48550/arxiv.1902.01983

Cited by 4 publications

(5 citation statements)

References 40 publications

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“…We thus hope that also from this point of view, our work will be of independent interest as we expect that it can be generalized to other probabilistic models where log-correlated fields arise. In particular, some of the results of Section 3 have already been used in [66,67].…”

Section: Main Results On Gaussian Multiplicative Chaosmentioning

confidence: 99%

“…[46] for original conjectures, [69,84,73] for related studies in the setting of random unitary matrices and the sine process, and [9] for a connection between the absolute value of the characteristic polynomial of random Hermitian matrices and multiplicative chaos. See also [22,17,66,67] for further recent studies connecting multiplicative chaos to random matrix theory and β-ensembles.…”

Section: Main Results On Gaussian Multiplicative Chaosmentioning

confidence: 99%

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

et al. 2021

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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 processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

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Section: Main Results On Gaussian Multiplicative Chaosmentioning

confidence: 99%

Section: Main Results On Gaussian Multiplicative Chaosmentioning

confidence: 99%

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

et al. 2021

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“…with the convention Im log(x + i∞) = π 2 . In contrast to the maximum of the random field for the imaginary part of log characteristic polynomial stated above, more research has been devoted to understanding the maximum of the random field for the real part of log characteristic polynomial of random matrices [3,6,16,31,15,18,27,8,25]. But none of these works are on generally distributed Wigner matrices, and also no rigorous result on the limiting distribution (fluctuation) is obtained, although conjectures have been made in [15,16,18] for CUE and GUE.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Cramér-von Mises statistic for the spectral distribution of random matrices

Bao

2019

Preprint

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Let F N and F be the empirical and limiting spectral distributions of an N × N Wigner matrix. The Cramér-von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between F N and F in ℓ 2 -norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE.

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“…The second expectation in (2.38) is the Laplace transform of smooth linear statistics, so that the loop equations techniques apply to prove it is of polynomial order, see [40,Theorem 1.3]. More precisely, [40] applies to the smooth function (1 − χ z1,δ ) log instead of (1 − χ z1,δ ) log N , but we can decompose log…”

Section: 25)mentioning

confidence: 99%

The distribution of overlaps between eigenvectors of Ginibre matrices

Bourgade

Dubach

2019

Probab. Theory Relat. Fields

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We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove:(i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables;(ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues;(iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale;(v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

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Maximum of the characteristic polynomial of the Ginibre ensemble

Cited by 4 publications

References 40 publications

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

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

On Cramér-von Mises statistic for the spectral distribution of random matrices

The distribution of overlaps between eigenvectors of Ginibre matrices

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