Local inhomogeneous circular law

Alt, Johannes; Erdős, László; Krüger, Torben

doi:10.1214/17-aap1302

Cited by 40 publications

(64 citation statements)

References 36 publications

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“…This shows that M ζ 0 (0) in fact agrees with the extension to the real line of the unique solution with positive imaginary part to (2.10). Alternatively, the analysis given in [8,14] could also be used to show this is the correct solution in the limit as ℑ(z) → 0. We now introduce some notation and conventions.…”

Section: Exact Solution Atmentioning

confidence: 99%

“…case, a rigorous proof of the convergence of the ESD was given by Bai in [11], under certain technical assumptions. This work was recently extended in [8] and [14] to include local spectral scales and very general variance profiles, respectively. The ESD of such matrices converges to a deterministic measure which is radially symmetric around origin and supported on a disk with radius given by the square root of the spectral radius of the variance profile.…”

Section: Introductionmentioning

confidence: 99%

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Power Law Decay for Systems of Randomly Coupled Differential Equations

Erdős¹,

Krüger²,

Renfrew³

2018

SIAM J. Math. Anal.

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We consider large random matrices X with centered, independent entries but possibly different variances. We compute the normalized trace of f (X)g(X * ) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled differential equations with random coefficients. We show that when the coupling is critical the norm squared of the solution decays like t −1/2 .

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Section: Exact Solution Atmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Power Law Decay for Systems of Randomly Coupled Differential Equations

Erdős¹,

Krüger²,

Renfrew³

2018

SIAM J. Math. Anal.

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“…In [47] it was proved under similar assumptions by means of the so-called fourth moment theorem, which requires that the first four moments of X jk match the corresponding moments of the standard Gaussian distribution. We also refer to the recent results [4] and [50]. The general case of m ≥ 1 was proved by Y. Nemish [37] who obtained a local version of Theorem 1.2 under sub-exponential assumptions.…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

“…Unfortunately, this requires to assume high finite moments of matrix entries. Another way is to assume that the matrix entries have absolutely continuous and bounded densities, see [4].…”

Section: Resultsmentioning

confidence: 99%

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Local laws for non-Hermitian random matrices and their products

Götze

Naumov

Tikhomirov

2019

Random Matrices: Theory Appl.

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The aim of this paper is to prove a local version of the circular law for non-Hermitian random matrices and its generalization to the product of non-Hermitian random matrices under weak moment conditions. More precisely we assume that the entries X (q) jk of non-Hermitian random matricesIt is shown that the local law holds on the optimal scale n −1+2a , 0 < a < 1/2, up to some logarithmic factor. We further develop a Stein type method to estimate the perturbation of the equations for the Stieltjes transform of the limiting distribution. We also generalize the recent results [8], [47] and [37]. and let A(·) be the Lebesgue measure on C. By w −→ we denote weak convergence of probability measures. We first assume that m = 1. Then the following result is the well-known circular law.Theorem 1.1 (Macroscopic circular law). Let X jk , 1 ≤ j, k ≤ n be i.i.d. complex r.v. with E X jk = 0, E |X jk | 2 = 1. Then µ n w −→ µ (1) a.s. as n tends to infinity, where dµ (1) (z) = p (1) (z)dA(z).

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Central Limit Theorem for Linear Eigenvalue Statistics of Non‐Hermitian Random Matrices

Cipolloni

Erdős

Schröder

2021

Comm Pure Appl Math

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We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2 C derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82], [56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32].

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Local inhomogeneous circular law

Cited by 40 publications

References 36 publications

Power Law Decay for Systems of Randomly Coupled Differential Equations

Power Law Decay for Systems of Randomly Coupled Differential Equations

Local laws for non-Hermitian random matrices and their products

Central Limit Theorem for Linear Eigenvalue Statistics of Non‐Hermitian Random Matrices

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