Optimal multi-resolvent local laws for Wigner matrices
Giorgio Cipolloni, L\'aszl\'o Erd\H{o}s, Dominik Schr\"oder
TL;DR
This paper establishes optimal local laws for products of resolvents of Wigner matrices with deterministic matrices, revealing how tracelessness affects concentration estimates at the smallest spectral scales.
Contribution
It introduces a comprehensive framework for local laws involving arbitrary resolvent products, explicitly incorporating the tracelessness condition of deterministic matrices.
Findings
Resolves optimal concentration estimates for resolvent products.
Shows the impact of traceless matrices on spectral scale estimates.
Provides bounds that are sharp at the smallest spectral scales.
Abstract
We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale.
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