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Bulk universality of general $$\beta$$-ensembles with non-convex potential. (English) Zbl 1278.82032
Summary: 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 the earlier work [the authors, “Universality of general $$\beta$$-ensembles”, preprint arxiv:0907.5605 (2011)]. 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.