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

Claeys, Tom; Fahs, Benjamin; Lambert, Gaultier; Webb, Christian

doi:10.1215/00127094-2020-0070

Cited by 35 publications

(34 citation statements)

References 93 publications

(188 reference statements)

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“…A similar result should hold for Im L N . Such a convergence has been shown for the CUE [82,69], unitarily invariant Hermitian random matrices [14,27], classical compact groups [35], and the GOE, GSE [48].…”

Section: Related Workmentioning

confidence: 74%

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Optimal local law and central limit theorem for $β$-ensembles

Bourgade,

Mody,

Pain

2021

Preprint

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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 following consequences. (i) The optimal rigidity scale of the ordered particles is of order (log N )/N in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scaleThe logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem.Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian β-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.

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Section: Related Workmentioning

confidence: 74%

“…The logarithmic factor in these rigidity bounds is optimal. Indeed, [27] shows that when β = 2, eigenvalues in the bulk can fluctuate from their expected locations by as much as c(log N )/N . Below and in the remainder of this paper, we denote k = min(k, N + 1 − k).…”

Section: Remark 12 Proposition 35 Extends This Local Law To E /mentioning

confidence: 99%

Optimal local law and central limit theorem for $β$-ensembles

Bourgade,

Mody,

Pain

2021

Preprint

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“…This fact follows e.g. from the construction of the random measure G in[4] -see also[11, Proposition 2.1] for further details.…”

mentioning

confidence: 85%

“…The following result follows essentially from [11,Proposition 3.8]. Since our context is slightly different, we provide the main steps of the proof for completeness.…”

Section: Thick Points: Proofs Of Proposition 16 and Proposition 17mentioning

confidence: 98%

“…Based on the approach from Berestycki [4], a general construction scheme which covers the whole subcritical regime was given in [34] and then refined in our recent work [11]. This method has been applied to (unitary invariant) Hermitian random matrices [11], as well as to the characteristic polynomial of the Ginibre ensemble [33]. A similar approach has also been applied to study the Riemann function [41] and cover times of planar Brownian motion [25].…”

Section: Subcritical Gaussian Multiplicative Chaosmentioning

confidence: 99%

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Mesoscopic central limit theorem for the circular beta-ensembles and applications

Lambert¹

2019

Preprint

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We give a simple proof of a central limit theorem for linear statistics of the Circular -ensembles which is valid at almost arbitrary mesoscopic scale and for functions of class C 3 . As a consequence, using a coupling introduced by Valkò and Viràg [46], we deduce a central limit theorem for the Sine processes. We also discuss the connection between our result and the theory of Gaussian Multiplicative Chaos. Based on the result of [34], we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the Circular -ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This implies that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions of log-correlated Gaussian fields.

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The maximum of log‐correlated Gaussian fields in random environment

Schweiger,

Zeitouni

2023

Comm Pure Appl Math

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We study the distribution of the maximum of a large class of Gaussian fields indexed by a box and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding et al., we show that asymptotically, the centered maximum of the field has a randomly‐shifted Gumbel distribution. We prove that the two dimensional Gaussian free field on a super‐critical bond percolation cluster with p close enough to 1, as well as the Gaussian free field in i.i.d. bounded conductances, fall under the assumptions of our general theorem.

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

Cited by 35 publications

References 93 publications

Optimal local law and central limit theorem for $β$-ensembles

Optimal local law and central limit theorem for $β$-ensembles

Mesoscopic central limit theorem for the circular beta-ensembles and applications

The maximum of log‐correlated Gaussian fields in random environment

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