Freeness over the diagonal for large random matrices

Au, Benson, Cébron, Guillaume, Dahlqvist, Antoine, Gabriel, Franck and Male, Camille (2021) Freeness over the diagonal for large random matrices. Annals of Probability, 49 (1). pp. 157-179. ISSN 0091-1798

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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.

Item Type: Article
Keywords: Random matrices, Permutation invariance, Free Probability, Amalgamation
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Subjects: Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
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Depositing User: Antoine Dahlqvist
Date Deposited: 12 May 2020 13:40
Last Modified: 22 Mar 2021 14:15

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