Local law for the product of independent non-Hermitian random matrices with independent entries

Nemish, Yuriy

doi:10.1214/17-ejp38

Cited by 21 publications

(25 citation statements)

References 20 publications

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“…With Lemma B.1 in hand, we are now prepared to prove Theorem 8.1. The proof below is based on a slight modification to the arguments from [47,48]. As such, in some places we will omit technical computations and only provide appropriate references and necessary changes to results from [47,48].…”

Section: Proofs Of Results From Sectionmentioning

confidence: 99%

Outliers in the spectrum for products of independent random matrices

Coston¹,

O’Rourke²,

Wood³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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For fixed m ≥ 1, we consider the product of m independent n × n random matrices with iid entries as n → ∞. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Tao [60] for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. 42 12. Proofs of results from Section 3 54 Appendix

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Section: Proofs Of Results From Sectionmentioning

confidence: 99%

Outliers in the spectrum for products of independent random matrices

Coston¹,

O’Rourke²,

Wood³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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“…In Local Laws [31,18], locally shrinking functions f z0 have been considered and for fixed global f Gaussian fluctuation has been proven in [24]. As was done in [16], we would like to uniformly approximate all indicator functions 1 B R (z0) by smooth functions, replace the right hand side of (3.8) by a discrete random sum and use the pointwise estimate from Proposition 3.1.…”

Section: Matrices With Independent Entriesmentioning

confidence: 99%

“…is the m-th power of the uniform distribution µ ∞ = µ 1 ∞ on the complex disc, see also [33]. The Gaussian case has been treated in [10,2], more general models can be found in [25,17,7,1,22], for the convergence of the singular values see [5] and furthermore for local results we refer to [31,24,32,18,12].…”

Section: Introductionmentioning

confidence: 99%

Rate of convergence for products of independent non-Hermitian random matrices

Jalowy

2021

Electron. J. Probab.

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We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by O(1/ √ n), which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.

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“…for i = 1, 2, it is often useful to let ϕ approximate an indicator function. In particular, allowing ϕ to depend on n, one can use Theorem 1.2 (along with the explicit formula for C 1.2 given in (2.15)) to obtain local laws which describe the mesoscopic behavior of the eigenvalues; such local laws have been established in the random matrix theory literature for a variety of ensembles, see [1,2,3,6,11,13,19,20,26,27,28,30,36,37,40,43,45,47,49,51,67,69,75,76,77] and references therein for a partial list of such results. We will use Theorem 1.2 to establish a rate of convergence for the empirical spectral measure of Toeplitz matrices subject to small random perturbations in Section 1.2.…”

Section: Introductionmentioning

confidence: 99%

Quantitative results for non-normal matrices subject to random and deterministic perturbations

O’Rourke¹,

Wood²

2021

Preprint

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We consider the eigenvalue distribution of a fixed matrix subject to a small perturbation. In particular, we prove quantitative comparison results which show that the logarithmic potential is stable under perturbations with small norm or low rank, provided the smallest and largest singular values are not too extreme. We also establish a quantitative version of the Tao-Vu replacement principle. As an application of our results, we study the spectral distribution of banded Toeplitz matrices subject to small random perturbations, including the case when the bandwidth grows with the dimension. For this model, we obtain a rate of convergence in Wasserstein distance of the empirical spectral measure to its limiting distribution. We give a number of examples of our methods including random multiplicative perturbations and large classes of deterministic perturbations that behave similar to random perturbations.

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Local law for the product of independent non-Hermitian random matrices with independent entries

Cited by 21 publications

References 20 publications

Outliers in the spectrum for products of independent random matrices

Outliers in the spectrum for products of independent random matrices

Rate of convergence for products of independent non-Hermitian random matrices

Quantitative results for non-normal matrices subject to random and deterministic perturbations

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