Large permutation invariant random matrices are asymptotically free over the diagonal

Abstract

We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, 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 (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erdős-Rényi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).

arXiv comment18 pages, 3 figures

References in corpus (4)

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• Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes
• A combinatorial theory of random matrices III: random walks on $\mathfrak{S}(N)$, ramified coverings and the $\mathfrak{S}(\infty )$ Yang-Mills measure
• Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit

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