Local laws for polynomials of Wigner matrices

Erdos L, Krueger T, Nemish Y (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Journal of Functional Analysis Elsevier

Book Volume: 278

Article Number: 108507

Journal Issue: 12

DOI: 10.1016/j.jfa.2020.108507

Abstract

We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.

Authors with CRIS profile

Torben Krüger

Involved external institutions

IST Austria AT Austria (AT) Rheinische Friedrich-Wilhelms-Universität Bonn DE Germany (DE)

How to cite

APA:

Erdos, L., Krueger, T., & Nemish, Y. (2020). Local laws for polynomials of Wigner matrices. Journal of Functional Analysis, 278(12). https://doi.org/10.1016/j.jfa.2020.108507

MLA:

Erdos, Laszlo, Torben Krueger, and Yuriy Nemish. "Local laws for polynomials of Wigner matrices." Journal of Functional Analysis 278.12 (2020).

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