Random Matrices with Slow Correlation Decay
L\'aszl\'o Erd\H{o}s, Torben Kr\"uger, Dominik Schr\"oder
TL;DR
This paper establishes universality and optimal local laws for large random matrices with slowly decaying correlations among entries, extending previous results to more general correlation structures.
Contribution
It introduces a systematic diagrammatic method for controlling multivariate cumulant expansions in matrices with slow correlation decay.
Findings
Proves universality of local eigenvalue statistics
Establishes optimal local laws for the resolvent
Generalizes previous results to broader correlation decay conditions
Abstract
We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of [arXiv:1604.08188] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
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