Optimal Local Law and Central Limit Theorem for β-Ensembles
Abstract
In the setting of generic β-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order (logN)/N in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale logN/N. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian β-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- March 2022
- DOI:
- arXiv:
- arXiv:2103.06841
- Bibcode:
- 2022CMaPh.390.1017B
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics
- E-Print:
- 47 pages, to appear in Comm. Math. Phys