Local inhomogeneous circular law

Alt J, Erdos L, Krueger T (2018)

Publication Type: Journal article, Review article

Publication year: 2018


Book Volume: 28

Pages Range: 148-203

Journal Issue: 1

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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Alt, J., Erdos, L., & Krueger, T. (2018). Local inhomogeneous circular law. Annals of Applied Probability, 28(1), 148-203. https://doi.org/10.1214/17-AAP1302


Alt, Johannes, Laszlo Erdos, and Torben Krueger. "Local inhomogeneous circular law." Annals of Applied Probability 28.1 (2018): 148-203.

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