  We prove the bulk universality of the $\beta$-ensembles with non-convex
regular analytic potentials for any $\beta>0$. 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.
