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Article Dans Une Revue Communications in Mathematical Physics Année : 2022

Optimal Local Law and Central Limit Theorem for $\beta $-Ensembles

Résumé

In the setting of generic $\beta$-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 $(\log N)/N$ 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 $\sqrt{\log N}/N$. (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 $\beta$-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.
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Dates et versions

hal-04281767 , version 1 (13-11-2023)

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Paul Bourgade, Krishnan Mody, Michel Pain. Optimal Local Law and Central Limit Theorem for $\beta $-Ensembles. Communications in Mathematical Physics, 2022, 390 (3), pp.1017-1079. ⟨10.1007/s00220-022-04311-2⟩. ⟨hal-04281767⟩
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