Mesoscopic linear statistics of Wigner matrices

Lodhia, Asad; Simm, Nick

doi:10.48550/arxiv.1503.03533

Cited by 20 publications

(22 citation statements)

References 57 publications

(89 reference statements)

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“…It allows us to prove a mesoscopic central limit theorem near the edge. The mesoscopic central limit at the bulk for Wigner matrices was proven in [6,7,25,35], for β-ensemble in [35] and for Dyson Brownian motion in [9,27,28]. As far as we know, the mesoscopic central limit theorem near the edge is new even for the Wigner matrices and β-ensembles.…”

Section: Introductionmentioning

confidence: 97%

Dyson Brownian Motion for General $β$ and Potential at the Edge

Adhikari

Huang

2018

Preprint

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In this paper, we compare the solutions of Dyson Brownian motion with general β and potential V and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and potential V , we obtain the optimal rigidity estimates of particle locations near the edge for short time t = o(1). Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge which, as far as we know, have been done for the first time in this paper. Additionally, combining with [30], our rigidity estimates are used to give a proof of the local ergodicity of Dyson Brownian motion for general β and potential at the edge, i.e. the distribution of extreme particles converges to Tracy-Widom β distribution in short time.

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Section: Introductionmentioning

confidence: 97%

Dyson Brownian Motion for General $β$ and Potential at the Edge

Adhikari

Huang

2018

Preprint

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“…Because of the rescaling by N α , the sums (1.3) typically involve about N 1−α eigenvalues, and consequently such linear statistics are refered to as mesoscopic. This result has been known in restricted cases for some time (see [10,11,24]). In [20], the authors obtain the result for general Wigner matrices and general f .…”

Section: Mesoscopic Linear Statisticsmentioning

confidence: 63%

Applications of mesoscopic CLTs in random matrix theory

Landon¹,

Sosoe²

2018

Preprint

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We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and β-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical β-ensembles (β = 1, 2, 4), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order o(N −1 ) for the expected value of eigenvalues in the bulk of the spectrum. The latter result solves a conjecture of Tao and Vu.

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“…However, special attention must be paid to the type of function f that arises from the homogenization theory; up to lower order corrections it is a smoothed out step function. In particular it is not of compact support, lying outside works such as [37,47]. The work [42] overcomes the limitations of previous approaches to linear spectral statistics by proving a mesoscopic CLT for test functions of noncompact support; the fluctuations for such mesoscopic statistics turn out to be on the scale a logpN q with the non-universal contributions to the variance again being removed by the extra a logpN q fac-tor.…”

Section: Proof Strategymentioning

confidence: 92%

“…Therefore, the second ingredient of the proof in [42] is a central limit theorem for mesoscopic LSS. Mesoscopic central limit theorems for functions f of compact support are well studied [37,47], and results analogous to (1.9). Note that when f is of compact support, the non-universal contributions to V pϕq vanish.…”

Section: Proof Strategymentioning

confidence: 99%

“…Our approach to linear spectral statistics via resolvent methods is most closely related to the work [42], which in turn was inspired by the approach using Stein's method of Shcherbina [55]. Fluctuations for mesoscopic linear spectral statistics of Wigner matrices have been obtained by Lodhia-Simm [47] on scales Imrzs " N ´1{3 and by He-Knowles [37] on scales Imrzs " N ´1. Further results in the Wigner case have been obtained by Bao-He [8], as well as Cipolloni-Erdős-Schröder [24].…”

Section: Relation To Prior Workmentioning

confidence: 99%

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Single eigenvalue fluctuations of general Wigner-type matrices

Landon¹,

Lopatto²,

Sosoe³

2021

Preprint

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We consider the single eigenvalue fluctuations of random matrices of general Wignertype, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance which agrees with the GOE and GUE cases. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain classes of test functions, the leading order contribution to the variance is universal, agreeing with the GOE/GUE cases.

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Mesoscopic linear statistics of Wigner matrices

Cited by 20 publications

References 57 publications

Dyson Brownian Motion for General $β$ and Potential at the Edge

Dyson Brownian Motion for General $β$ and Potential at the Edge

Applications of mesoscopic CLTs in random matrix theory

Single eigenvalue fluctuations of general Wigner-type matrices

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