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Bulk Universality of General $β$-Ensembles with Non-convex Potential

We prove the bulk universality of the $β$-ensembles with non-convex regular analytic potentials for any $β>0$. This removes the convexity assumption appeared in our earlier work. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a "convexified measure" so that the local statistics are preserved under this convexification.