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February 2018 Local inhomogeneous circular law
Johannes Alt, László Erdős, Torben Krüger
Ann. Appl. Probab. 28(1): 148-203 (February 2018). DOI: 10.1214/17-AAP1302


We consider large random matrices $X$ with centered, independent entries, which have comparable but not necessarily identical variances. Girko’s circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et al. [Probab. Theory Related Fields 159 (2014) 545–595; Probab. Theory Related Fields 159 (2014) 619–660] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of $X$.


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Johannes Alt. László Erdős. Torben Krüger. "Local inhomogeneous circular law." Ann. Appl. Probab. 28 (1) 148 - 203, February 2018.


Received: 1 February 2017; Revised: 1 April 2017; Published: February 2018
First available in Project Euclid: 3 March 2018

zbMATH: 06873682
MathSciNet: MR3770875
Digital Object Identifier: 10.1214/17-AAP1302

Primary: 60B20
Secondary: 15B52

Keywords: circular law , Local law , variance profile

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 1 • February 2018
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