Edge universality for non-Hermitian random matrices
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Edge universality for non-Hermitian random matrices Giorgio Cipolloni1 · László Erdos ˝ 1
· Dominik Schröder2
Received: 9 September 2019 / Revised: 9 September 2020 / Accepted: 13 September 2020 © The Author(s) 2020
Abstract We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble. Keywords Ginibre ensemble · Universality · Circular law · Girko’s formula Mathematics Subject Classification 60B20 · 15B52
1 Introduction Following Wigner’s motivation from physics, most universality results on the local eigenvalue statistics for large random matrices concern the Hermitian case. In particular, the celebrated Wigner–Dyson statistics in the bulk spectrum [44], the Tracy– Widom statistics [56,57] at the spectral edge and the Pearcey statistics [47,58] at the possible cusps of the eigenvalue density profile all describe eigenvalue statistics of a large Hermitian random matrix. In the last decade there has been a spectacular progress in verifying Wigner’s original vision, formalized as the Wigner–
Partially supported by ERC Advanced Grant No. 338804 and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385.
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László Erd˝os [email protected] Giorgio Cipolloni [email protected] Dominik Schröder [email protected]
1
IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
2
Institute for Theoretical Studies, ETH Zurich, Clausiusstr. 47, 8092 Zurich, Switzerland
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G. Cipolloni et al.
Dyson–Mehta conjecture, for Hermitian ensembles with increasing generality, see e.g. [2,15,23–26,35,37,40,42,45,48,52,52] for the bulk, [5,12,13,34,38,39,46,50,53] for the edge and more recently [17,22,33] at the cusps. Much less is known about the spectral universality for non-Hermitian models. In the simplest case of the Ginibre ensemble, i.e. random matrices with i.i.d. standard Gaussian entries without any symmetry condition, explicit formulas for all correlation functions have been computed first for the complex case [31] and later for the more complicated real case [10,36,49] (with special cases solved earlier [20,21,43]). Beyond the explicitly computable Ginibre case only the method of four moment matching by Tao and Vu has been available. Their main universality result in [54] states that the local correlation functions of the eigenvalues of a random matrix X with i.i.d. matrix elements coincide with those of the Ginibre ensemble as long as the first four moments of the common distribution of the entries of X (almost) match the first four moments of the standard Gaussian. This result holds for both real and complex cases as well as throughout the spectrum, incl
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