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January 2021 Freeness over the diagonal for large random matrices
Benson Au, Guillaume Cébron, Antoine Dahlqvist, Franck Gabriel, Camille Male
Ann. Probab. 49(1): 157-179 (January 2021). DOI: 10.1214/20-AOP1447

Abstract

We prove that independent families of permutation invariant random matrices are asymptotically free with amalgamation over the diagonal, both in expectation and in probability, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (e.g., to Wigner matrices with exploding moments and the sparse regime of the Erdős–Rényi model). The result still holds even if the matrices are multiplied entrywise by random variables satisfying a certain growth condition (e.g., as in the case of matrices with a variance profile and percolation models). Our analysis relies on a modified method of moments based on graph observables.

Citation

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Benson Au. Guillaume Cébron. Antoine Dahlqvist. Franck Gabriel. Camille Male. "Freeness over the diagonal for large random matrices." Ann. Probab. 49 (1) 157 - 179, January 2021. https://doi.org/10.1214/20-AOP1447

Information

Received: 1 September 2019; Revised: 1 March 2020; Published: January 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1214/20-AOP1447

Subjects:
Primary: 15B52 , 46L54
Secondary: 46L53 , 60B20

Keywords: freeness with amalgamation , permutation invariance , random matrices , traffic probability

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 1 • January 2021
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