In the setting of generic $\beta$-ensembles, we use loop equation hierarchy to prove a local law with optimal error up constant, valid on any scale including microscopic. This has following consequences. (i) The rigidity ordered particles is order $(\log N)/N$ in bulk spectrum. (ii) Fluctuations satisfy central limit theorem covariance corresponding logarithmically correlated field; particular ...

Abstract We study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in high-temperature regime this central limit theorem holds for vector suitably scaled group magnetisations, that is, sum spins belonging to each group. In article, we prove local magnetisations regime.