Local single ring theorem

Benaych-Georges, Florent

doi:10.1214/16-aop1151

Cited by 4 publications

(7 citation statements)

References 30 publications

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“…Since the typical distance between the eigenvalues in the bulk of the ring R σ is of order N −1/2 , our result is optimal, both in terms of range of the exponent α and the error term on the right side of (1.19). In particular, this improves the recent local single ring theorem of Benaych-Georges in [10] from scale (log N ) −1/4 to the optimal scale N −1/2+ǫ , for any small ǫ > 0. Remark 1.11.…”

Section: )supporting

confidence: 70%

“…Denote by m 1 ⊕ m 1 the Lie algebra of M 1 × M 1 . The following argument is essential due to [33]; see also [10,28] for similar arguments. Viewing the Green function as a function (random variable) on M × M, G(·, z) : M × M → M N (C), we compute, using (8.4), that…”

Section: Proof Of Theorem 43 For Large ηmentioning

confidence: 95%

“…With this strategy, the local circular law on optimal scale was established in the series of works Bourgade, Yau and Yin [14,15] and Yin [38]. The first local single ring theorem was obtained by Benaych-Georges in [10], down to the scale (log N ) − 1 4 , by proving the matrix subordination for Girko's Hermitization of X in (1.1), c.f., (2.14). The strategy of matrix subordination was originally introduced by Kargin in [28] for proving a local law in the additive matrix model A + U BU * , where A and B are deterministic Hermitian matrices and U is a Haar unitary.…”

Section: )mentioning

confidence: 99%

“…It builds on the celebrated Gromov-Milman concentration inequality whose application to random matrix theory is fairly standard [1]. For additive models of the form X + U Y U * , with deterministic X, Y ∈ M N (C) and U Haar distributed on U (N ) or on O(N ) it was used in [33,28,2], and for the model block-additive model considered in this section in [10].…”

Section: 2mentioning

confidence: 99%

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Local single ring theorem on optimal scale

Bao¹,

Erdős²,

Schnelli³

2019

Ann. Probab.

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Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U (N ). Let Σ be a non-negative deterministic N by N matrix. The single ring theorem [26] asserts that the empirical eigenvalue distribution of the matrix X := U ΣV * converges weakly, in the limit of large N , to a deterministic measure which is supported on a single ring centered at the origin in C. Within the bulk regime, i.e., in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N −1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N ).Remark 1.3. In Theorem 1.2, U and V may be both Haar distributed on U (N ) or on O(N ). Remark 1.4. In its original form Theorem 1.2 was proved by Guionnet, Krishnapur and Zeitouni in [26] under a further assumption on the smallest singular value of the matrix X − z, z ∈ C. This hard-to-check condition was removed by Rudelson and Vershynin in [34] (c.f., Theorem 2.6 below), which yields Theorem 1.2.Remark 1.5. The measure ρ σ was first computed in [27]. It has a direct interpretation in free probability theory, in fact it is the Brown measure of the free product of a Haar unitary and an element σ on a noncommutative probability space; see [27] for more details. * We will often use the convention that capital letters indicate random matrices and the corresponding small letters indicate their limiting objects.

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Section: )supporting

confidence: 70%

Section: Proof Of Theorem 43 For Large ηmentioning

confidence: 95%

Section: )mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Local single ring theorem on optimal scale

Bao¹,

Erdős²,

Schnelli³

2019

Ann. Probab.

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“…Via the hermitization technique, local laws for the addition of random matrices can be used to prove local versions of the single ring theorem. This approach was demonstrated recently by Benaych-Georges [8], who proved a local single ring theorem on scale (log N ) −1/4 using Kargin's local law on scale (log N ) −1/2 . The local law on the optimal scale N −1 is one of the key ingredients to prove the local single ring theorem on the optimal scale.…”

Section: Introductionmentioning

confidence: 93%

Local Law of Addition of Random Matrices on Optimal Scale

Bao

Erdős

Schnelli

2016

Commun. Math. Phys.

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Abstract:The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.

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Anisotropic local laws for random matrices

Knowles

Yin

2016

Probab. Theory Relat. Fields

137

221

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We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity. For definiteness, we present the method for sample covariance matrices of the form Q . .= T XX * T * , where T is deterministic and X is random with independent entries. We prove that with high probability the resolvent of Q is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales.As an application, we prove the edge universality of Q by establishing the Tracy-Widom-Airy statistics of the eigenvalues of Q near the soft edges. This result applies in the single-cut and multi-cut cases. Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere.We also apply our method to Wigner matrices whose entries have arbitrary expectation, i.e. we consider W + A where W is a Wigner matrix and A a Hermitian deterministic matrix. We prove the anisotropic local law for W + A and use it to establish edge universality.

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Local single ring theorem

Cited by 4 publications

References 30 publications

Local single ring theorem on optimal scale

Local single ring theorem on optimal scale

Local Law of Addition of Random Matrices on Optimal Scale

Anisotropic local laws for random matrices

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