Optimal lower bound on the least singular value of the shifted Ginibre ensemble

Cipolloni, Giorgio; Erdős, László; Schröder, Dominik

doi:10.2140/pmp.2020.1.101

Cited by 17 publications

(11 citation statements)

References 60 publications

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“…where ρ locally vanishes as a square-root. According to[24, Eq. (15b)] the density EJP 26 (2021), paper 24. ρ has two regular edges ± √ e + if |z| ≤ 1 − , and four regular edges in ± √ e + , ± √ e − for |z| ≥ 1 + , where By the explicit form of e ± it follows that e ± 1 whenever |1 − |z|| ≥ .…”

mentioning

confidence: 99%

Fluctuation around the circular law for random matrices with real entries

Cipolloni

Erdős

Schröder

2021

Electron. J. Probab.

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We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59,44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

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confidence: 99%

Fluctuation around the circular law for random matrices with real entries

Cipolloni

Erdős

Schröder

2021

Electron. J. Probab.

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“…The edge regime is much less studied. However, as it was shown in [8] for the case of the constant diagonal shift of the Ginibre ensemble, i.e. A = −zI, the bound (1.2) can be improved in the edge regime |z| ∼ 1.…”

Section: Introductionmentioning

confidence: 70%

The least singular value of the general deformed Ginibre ensemble

Shcherbina¹,

Shcherbina²

2022

Preprint

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We study the least singular value of the n × n matrix H − z with H = A 0 + H 0 , where H 0 is drawn from the complex Ginibre ensemble of matrices with iid Gaussian entries, and A 0 is some general n × n matrix with complex entries (it can be random and in this case it is independent of H 0 ). Assuming some rather general assumptions on A 0 , we prove an optimal tail estimate on the least singular value in the regime where z is around the spectral edge of H thus generalize the recent result of Cipolloni, Erdős, Schröder [8] to the case A 0 = 0. The result improves the classical bound by Sankar, Spielman and Teng [21].

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“…at |z| = 1) in [16]. The last result strongly relies on an estimate for the least singular value obtained in [13] using the supersymmetry technique (SUSY).…”

Section: Introductionmentioning

confidence: 99%

“…There are also a lot of rigorous results, which were obtained using SUSY in the recent years, e.g. [6,13,[17][18][19][20][21][49][50][51][52] etc. Supersymmetry technique is usually used in order to obtain an integral representation for ratios of determinants.…”

Section: Introductionmentioning

confidence: 99%

On the Correlation Functions of the Characteristic Polynomials of Real Random Matrices with Independent Entries

Afanasiev¹

2020

Z. mat. fiz. anal. geom.

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The paper is concerned with the correlation functions of the characteristic polynomials of real random matrices with independent entries. The asymptotic behavior of the correlation functions is established in the form of a certain integral over unitary self-dual matrices with respect to the invariant measure. The integral is computed in the case of the second order correlation function. From the obtained asymptotics it is clear that the correlation functions behave like that for the Real Ginibre Ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries.

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Optimal lower bound on the least singular value of the shifted Ginibre ensemble

Cited by 17 publications

References 60 publications

Fluctuation around the circular law for random matrices with real entries

Fluctuation around the circular law for random matrices with real entries

The least singular value of the general deformed Ginibre ensemble

On the Correlation Functions of the Characteristic Polynomials of Real Random Matrices with Independent Entries

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