Edge universality for non-Hermitian random matrices

Cipolloni, Giorgio; Erdős, László; Schröder, Dominik

doi:10.1007/s00440-020-01003-7

Cited by 38 publications

(84 citation statements)

References 58 publications

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“…[4,Section 3.3]. Then the local law for any η > 0 readily follows by exactly the same argument as in [23,Appendix A]. This completes the proof of Theorem 3.1.…”

Section: Local Law Away From the Imaginary Axis: Proof Of Theorem 31supporting

confidence: 58%

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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“…[4,Section 3.3]. Then the local law for any η > 0 readily follows by exactly the same argument as in [23,Appendix A]. This completes the proof of Theorem 3.1.…”

Section: Local Law Away From the Imaginary Axis: Proof Of Theorem 31supporting

confidence: 58%

Fluctuation around the circular law for random matrices with real entries

Cipolloni

Erdős

Schröder

2021

Electron. J. Probab.

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“…We emphasize that the limit law (1.7) is not a feature of the GinOE alone. In fact, Cipolloni, Erdős and Schröder recently proved in [15,Theorem 2.3] that the edge eigenvalue statistics of a large class of real non-Hermitian random matrices with i.i.d. centered entries match those of the GinOE.…”

Section: Riemann-hilbertmentioning

confidence: 99%

Edge distribution of thinned real eigenvalues in the real Ginibre ensemble

Baik¹,

Bothner²

2020

Preprint

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This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed independently with constant likelihood. We show that the recently discovered integrable structures in [2] generalize from the real Ginibre ensemble to its thinned equivalent. Concretely, we express the aforementioned limiting distribution function as a convex combination of two simple Fredholm determinants and connect the same function to the inverse scattering theory of the Zakharov-Shabat system. As corollaries, we provide a Zakharov-Shabat evaluation of the ensemble's real eigenvalue generating function and obtain precise control over the limiting distribution function's tails. The latter part includes the explicit computation of the usually difficult constant factors.

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“…The condition of matching moments was recently overcome at the edge of the spectrum (i.e. 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%

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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Edge universality for non-Hermitian random matrices

Cited by 38 publications

References 58 publications

Fluctuation around the circular law for random matrices with real entries

Fluctuation around the circular law for random matrices with real entries

Edge distribution of thinned real eigenvalues in the real Ginibre ensemble

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

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